To complement our previous analysis of interactions of large-scale turbulence with strong detonations, the corresponding theory of interactions of small-scale turbulence is presented here. Focusing most directly on the results of greatest interest, the ultimate long-time effects of high-frequency vortical and entropic disturbances on the burnt-gas flow, a normal-mode analysis is selected here, rather than the Laplace-transform approach. The interaction of the planar detonation with a monochromatic pattern of perturbations is addressed first, and then a Fourier superposition for two-dimensional and three-dimensional isotropic turbulent fields is employed to provide integral formulas for the amplification of the kinetic energy, enstrophy, and density fluctuations. Effects of the propagation Mach number and of the chemical heat release and the chemical reaction rate are identified, as well as the similarities and differences from the previous result for the thin-detonation (fast-reaction) limit.
In previous work,1 we analyzed the interaction of a strong detonation with inhomogeneous density fields. The motivation was to improve the descriptions of the influence of compressible turbulence on detonation propagation and, in particular, to determine how passage of a planar detonation modifies the turbulence. In that respect, the work complemented the analyses of Jackson et al.,2,3 who addressed interactions of detonations with constant-density vorticity fields. It is well known4 that, excluding acoustic perturbations, which propagate with respect to the fluid, inhomogeneities that travel with the fluid can be decomposed into vortical and entropic components. The earlier work2,3 had explored the influences of the vortical component, but at high Mach numbers those influences are dominated by influences of the entropic component,5 which therefore was in need of study. The previous investigation1 provided the additional information that was needed for strong detonations that could be treated as discontinuities.
General reasons for interest in interactions between detonations and turbulence have been discussed and referenced earlier,1 and, since the present paper relies on knowledge of the previous paper, those reasons will not be repeated here. One motivation is improvement in the performance of detonative propulsion devices for hypersonic aircraft, where limited times for fuel-air mixing can pose problems that could be attacked through shock-wave-enhanced mixing.6,7 In such situations, detonation thicknesses may be larger than the most important representative turbulence scales. The earlier analysis1 then become inaccurate, and turbulence effects on the evolution of the finite-rate heat release in the interior of the detonation become relevant. The present investigation offers a step towards removing this deficiency.
It is worth observing that numerical studies by Massa et al.8,9 have shown enhanced interactions between turbulence and detonations associated with there being comparable sizes of the inhomogeneities and the unperturbed reaction zones. This increases the interest in pursuing analytical investigations in which turbulence scales are not large compared with reaction-zone thicknesses, for example, for testing scaling hypotheses and to determine whether effects of systematic variations of parameters can be derived. Such investigations become quite difficult when the reaction zones and inhomogeneities are of comparable size, but they can be performed accurately when the dimensions of all disturbances are small compared with reaction-zone thicknesses. This is the limit to be addressed here. Inferences concerning behavior at intermediate scales may be derived from resulting functional dependences and comparisons with previous thin-detonation results. Since neither vortical nor entropic fluctuations have yet been considered in this limit, both will be analyzed in the present work.
II. THE STEADY, PLANAR DETONATION
The analysis is a perturbation of the steady, planar ZND detonation structure, a shock followed by an inviscid reaction zone,10–12 for an ideal gas with a constant specific heat at constant pressure. The formulation will be entirely in terms of nondimensional variables. For simplicity, we assume that the rate of energy release per unit volume behind the lead shock
Figure 1, in which D denotes the nondimensional propagation velocity of the detonation, illustrates the particle paths and velocity profiles in the laboratory frame of reference. The upstream conditions can be expressed in units of their corresponding Neumann values by means of the Rankine-Hugoniot equations, which lead to
The Neumann Mach number defined as the nondimensional velocity at which the particles leave the shock front is
For strong detonations, Mo is appreciably greater than its Chapman-Jouget value,
which is related to the velocity profile according to
III. FORMULATION OF THE PERTURBATION PROBLEM
Complex notation is employed in the normal-mode analysis, with the physical values represented by the real parts. The upstream vortical velocity field is defined in terms of the nondimensional longitudinal
where the divergence-free condition
where εe is the nondimensional amplitude, assumed to be small, like εr. Since the linear responses are independent, no phase angle is imposed between the rotational (subscript r) and entropic (subscript e) perturbations.
Incoming waves from behind the detonation are excluded. When the lead shock encounters the perturbations in (5) and (6), it develops a time-dependent response proportional to
For a given harmonic excitation, such as that considered in (5) and (6), the linear disturbances following the shock front also must be harmonic. At leading order, the perturbations are adiabatic so an acoustic wave form describes the post-shock perturbation field, the acoustic frequency and wave number being related to the shock frequency by the expression
The effect of the chemical reaction enters through the dimensionless rate of heat release
which is scaled with use made of the detonation thickness
There can be stable acoustic radiation right behind the shock wave if
θ being the angle between the incident wavenumber
plus terms of order
Equation (12) is the generalization of the well-known expression for the response of the pressure behind the shock wave, accounting for the exothermic finite-rate chemistry in the heat-release zone of the detonation that follows the shock. The present contribution analyzes the influences of the terms involving
Although the formulation leading to (12) is then independent of the heat-release processes occurring downstream, those processes do affect the disturbances that occur within the thick detonation and downstream therefrom. As is well known from the earlier investigations, those disturbances can be considered to be of two general types, namely, acoustic fluctuations that propagate downstream with respect to the fluid locally and fluctuations that remain fixed with respect to the fluid particles. In the present notation, the former are proportional to
for the acoustic component, and the corresponding acoustic frequency experienced by a fluid element moving with the local fluid speed is
The results to be given below pertain to the coefficients of these disturbances, in particular, as the fluctuations emerge into the burnt gas. There are contributions to theses modification associated with passage through the heat-release region at order ε as well as order
IV. RESULTS OF THE PERTURBATION ANALYSIS
In terms of the expansions in (12), the results are quite different, depending on whether the acoustics is radiating (ζ ⩾ 1) or evanescent (ζ < 1). For radiating conditions, there is no contribution at order
The aforementioned indirect effect of the heat release in the detonations on the acoustic perturbations at leading order do, however, influence the final radiation condition. As a consequence of the variation of the quantities M and a with the distance behind the shock, the condition for the detonation to be radiating at its downstream boundary is ζ > ζb, where
which is plotted in Fig. 2 as a function of f − 1 (where
Because of the dominance of the lead shock for thick detonations, the amplitudes of the perturbation at order ε are the same as those derived previously for shock-vorticity interactions (εe = 0)13,14 and for shock-density interactions (εr = 0).15 Since there are no perturbations at order
where the subscripts r and e identify the rotational and entropic contributions to ε, respectively. The factors Bpr and Bpe refer to the pressure disturbances generated by the inert shock (see the Appendix for details) and are given by
In a similar manner, if the contributions to the density and vorticity fluctuations in the burnt gas at order
The velocity field is also modified at order
V. PREDICTIONS FOR MONOCHROMATIC DISTURBANCES
In viewing the corrections proportional to
All the curves have certain attributes in common. For example, all of the corrections approach zero as 1 − ζ approaches zero and unity, which, according to (11), correspond, respectively, to the critical angle θ and to the detonation encountering the perturbation field edge-on. It is understandable that an edge-on encounter will not have any effect, and since the corrections are of higher order for the angles of encounter between normal and the critical angle (that is, at the higher frequencies), considerations of continuity suggest that the approach to zero at the critical angle is reasonable. In addition, all of the curves achieve both negative and positive values, although, for purely entropic initial perturbations, the range and extent of the negative segment are very small. The curves, in general, exhibit oscillations, in that they pass through zero once or twice. This complicates the task of drawing general conclusions.
Since the logarithmic scale in Fig. 3 accentuates the range as the critical angle is approached, in determining the dominant overall effect, the scale variation must be considered. With this in mind, it may be inferred from the curves that the corrections are mainly positive for entropic upstream disturbances and mainly negative for rotational upstream disturbances. It was found that the heat release in thin detonations tends to reduce fluctuations for entropic upstream fluctuations.1 Although the curves in the earlier paper2 at first glance suggest otherwise, in fact, with the current scaling, the same is true for rotational fluctuations.2 Hence, the present thick-detonation results are in opposite directions from the thin-detonation results for entropic upstream disturbances but in the same direction for rotational disturbances. This suggests that entropic disturbances, which tend to be dominant at high Mach numbers, may experience enhanced responses at intermediate sizes.
VI. STATISTICAL AVERAGES FOR INTERACTIONS WITH ISOTROPIC TURBULENCE
Statistical averages for turbulent flows are generated from the preceding results by integrating the square amplitudes of the perturbations over the spectrum of wave numbers. As in previous work,1,3 the upstream flow is assumed to be homogeneous and isotropic so that the wave-number vector
When considering the interaction with a two-dimensional isotropic vorticity field, the velocity field is represented by divergence-free velocity disturbances
Thanks to isotropy upstream, the spectrum contribution
in that case. If there are only density disturbances ahead of the detonation wave, then the statistical averages under the same isotropy assumptions are1,15,19
where εe(k) represents the normalized internal-energy spectrum. In all cases results are given as ratios of final to initial normalized intensities, independent of the particular spectral distributions.
Consistent with the monochromatic perturbations to the inert lead-shock results being proportional to
There are similar breakdowns for enstrophy and density. Here, too, to avoid introducing an additional dependence on
A. Turbulent kinetic-energy amplification
for two-dimensional and three-dimensional disturbances, respectively, where the functions
for the two-dimensional and three-dimensional cases, respectively. The factors
for both two and three dimensions, and
for two dimensions and three dimensions, respectively. The factor
As explained above, when the upstream perturbations are purely entropic the ratio K is normalized as in Ref. 1, which results in
for the two-dimensional and three-dimensional cases, where the corresponding normalized probability density functions are
Here K is unity when the integrand functions equal unity. Splitting (30) into longitudinal and transverse kinetic-energy contributions with the definition K = L + T for both two-dimensional and three-dimensional cases results at order
Since the normalization of the turbulent kinetic energy is different for upstream vorticity
B. Downstream enstrophy
The effect of the inert shock front on the enstrophy levels is given in (A15) and agrees with previous results in shock-turbulence interactions.14,18 The two-dimensional and three-dimensional contributions of order
for rotational disturbances. For entropic disturbances, there is generation of vorticity, and following the same reasoning as in (30), we normalize the enstrophy downstream as in Refs. 1 and 15, so the contribution for both two and three dimensions become
Figure 5 shows the enstrophy deviation
C. Downstream density
The average of the downstream square-density perturbations is denoted by
for two-dimensional and three-dimensional cases, respectively, and the breakdown
for both two and three dimensions. For purely entropic fluctuations, the contribution at order εq|k|−1 is
Figure 6 shows the associated average-density deviation factor
To complement previous analyses of interactions of detonation with turbulence that treated all turbulence scales as being large compared with the thickness of the detonation, in the present work, the turbulence scales were taken to be small compared with the thickness of the reaction zones that follow the lead inert shock in the detonation. Disturbances in the fresh mixture that involve only velocity fluctuations without any fluctuation of thermodynamic properties (rotational disturbances), and disturbances that involve fluctuations of the density of the fresh mixture without any velocity fluctuation (entropic disturbances) were considered separately, although results for cases in which both types of disturbances were present simultaneously are given in the Appendix. It was found that, unlike the previous results for thin detonations, where the jump conditions across the complete detonation determined how the initial fluctuations were modified, for these thick detonations the modifications are affected mainly by the jump conditions across the leading inert shock. The wavelengths, frequencies, and phases of the disturbances vary as they pass through the heat-release zones, but the amplitudes do not, to leading order in the perturbations. Moreover, the effect of the rate of heat release
A finding that was not unexpected is that the influences on monochromatic fluctuations are strongest when the interaction occurs near the critical angle of incidence that separates downstream disturbances behind the lead shock that are supersonic from those that are subsonic. This motivated exhibiting results in a variable stretched about the critical angle, to maximize the efficiency of presentation. The expansion parameter of the analysis
For purely rotational disturbances (those without fluctuations of thermodynamic properties) in the fresh mixture, the heat release in the detonation reduces the relative fluctuation intensities downstream below the levels that would exist behind the inert lead shock, in both thin-detonation and thick-detonation limits. But for purely entropic disturbances in the fresh mixture (those without any velocity fluctuation), the effects of the heat release are the opposite in the two limits, the intensities being decreased for thin detonations but increased for thick detonations. This clearly suggests that when density fluctuations are dominant over velocity fluctuations, a situation thought to arise in turbulent flows at high Mach numbers,5 the influence of the passage of the detonation on the fluctuation field is affected qualitatively by the ratio of the detonation thickness to the range of sizes of the initial fluctuations. Smaller-scale fluctuations may be amplified, while longer-scale fluctuations are damped. The associated enhancement of smaller-scale disturbances may improve mixing rates and thereby promote combustion.
This work was supported by the U.S. AFOSR Grant No. FA9550-12-1-0138.
APPENDIX: MORE DETAILED AND ADDITIONAL RESULTS
As indicated is Sec. IV, the base perturbations of order ε were obtained in previous work, namely, in Ref. 14 for rotational fluctuations and in Ref. 15 for entropic fluctuations. The formulas are different for short-wavelength (ζ ⩾ 1) and long-wavelength (ζ < 1) cases, and only the latter appears in Sec. IV. In addition, the base pressure perturbations are affected by a second long-wavelength contribution that is orthogonal to the contribution in Sec. IV and that therefore does not appear there. This second long-wavelength contribution will be identified by a prime, and the short-wavelength base perturbations will be identified by a double prime. Both of these influence later results, and therefore both are summarized here.
Addressing first the rotational contributions, it was found that14
The resulting amplitude of the asymptotic vorticity field generated by the upstream rotational mode is then
where Ω1 is the direct amplification contribution generated by compression effects, and Ω2 is the associated shock-curvature effect, here given by
The associated density perturbations are
For the entropic contributions it was found that15
The resulting the vorticity-disturbance coefficients are
where Ω3 is the baroclinic contribution (i.e., it accounts for the vorticity caused by the streamwise pressure jump across the shock and transverse density fluctuations) and is given by
The corresponding amplitudes of the steady density perturbations are
Besides velocity fluctuations associated with the vorticity fluctuations addressed here, there are also acoustic contributions to the velocity perturbations, which will be identified with the superscript a. For rotational disturbances, the associated amplitudes of the x and y components are
while the acoustic contributions are
The relationships in (A9a), written there for the long-wavelength cases, also apply for the short-wavelength cases. The corresponding contributions to the velocity field at order
where the factor Aωr is defined in (21). For entropic disturbances the amplitudes are
for the rotational and acoustic contributions, respectively. Similarly, the vortical contributions to the velocity field at order at order
where the factors Aωe are also provided in (21).
and the transverse
When the upstream flow consists of isotropic solenoidal perturbations, the amplification of those perturbations in two and three dimensions is, respectively,
where the three-dimensional contribution can be split into
and detailed study of the enstrophy generated by shock-entropy interaction may be found in Refs. 15 and 20. The inert base contribution to the square-density perturbations behind the detonation, defined by (35), is1
When there are both rotational and entropic fluctuations upstream, phase shifts introduce interferences that prevent results from being obtained by simply adding the separate contributions. Previous work21,22 has addressed the two cases