The interaction between a weakly turbulent free stream and a hypersonic shock wave is investigated theoretically by using linear interaction analysis (LIA). The formulation is developed in the limit in which the thickness of the thermochemical nonequilibrium region downstream of the shock, where relaxation toward vibrational and chemical equilibrium occurs, is assumed to be much smaller than the characteristic size of the shock wrinkles caused by turbulence. Modified Rankine–Hugoniot jump conditions that account for dissociation and vibrational excitation are derived and employed in a Fourier analysis of a shock interacting with three-dimensional isotropic vortical disturbances. This provides the modal structure of the post-shock gas arising from the interaction, along with integral formulas for the amplification of enstrophy, concentration variance, turbulent kinetic energy (TKE), and turbulence intensity across the shock. In addition to confirming known endothermic effects of dissociation and vibrational excitation in decreasing the mean post-shock temperature and velocity, these LIA results indicate that the enstrophy, anisotropy, intensity, and TKE of the fluctuations are much more amplified through the shock than in the thermochemically frozen case. In addition, the turbulent Reynolds number is amplified across the shock at hypersonic Mach numbers in the presence of dissociation and vibrational excitation, as opposed to the attenuation observed in the thermochemically frozen case. These results suggest that turbulence may persist and get augmented across hypersonic shock waves despite the high post-shock temperatures.

Strong shock waves participate in a number of problems in physics, including the dynamics of high-energy interstellar medium,1–4 the explosions of giant stars,5–8 the fusion of matter in inertial-confinement devices,9–11 and the ignition of combustible mixtures by lasers.12,13 In addition to those, an important contemporary problem of relevance for aeronautical and astronautical engineering is the aerothermodynamics of hypersonic flight.14,15 In hypersonics, similarly to the aforementioned problems, the intense compression of the gas through the shock waves generated by the fuselage leads to high temperatures that can activate complex thermochemical phenomena.16 In particular, at high Mach numbers of up to approximately 25 in the terrestrial atmosphere, corresponding to sub-ionizing, sub-orbital stagnation enthalpies of up to approximately 15–30 MJ/kg depending on altitude, vibrational excitation, and air dissociation are the dominant thermochemical phenomena typically observed in the gas downstream of shock waves around hypersonic flight systems.

Turbulence can also play an important role at the high Mach numbers mentioned above, particularly in low-altitude hypersonic flight because of the correspondingly larger Reynolds numbers of the airflow around the fuselage.17–19 However, the way in which turbulence influences the thermomechanical loads and the thermochemistry around hypersonic flight systems remains largely unknown. To compound this problem, experiments in the area of hypersonic turbulence are curtailed by the exceedingly large flow powers required to move gases at sufficiently high Mach and Reynolds numbers in order to observe shock waves simultaneously with turbulence and thermochemistry. In addition, the airflow in most ground facilities is poisoned with weak free-stream turbulence that interacts with the shock waves enveloping the test article. The fluctuations in the post-shock gases induced by this interaction oftentimes lead to artificial transition to turbulence in hypersonic boundary layers in wind tunnel experiments.20 

Most early work on the interaction of shock waves with turbulence has been limited to calorically perfect gases in boundary layers21–30 and isotropic free streams.31–35 Large-scale numerical simulations, including Direct Numerical Simulations (DNS),36–50 Large Eddy Simulations (LES),51–53 and Reynolds-Averaged Navier-Stokes Simulations (RANS),54,55 have been the pacing item for those investigations. Nonetheless, the rapid progress in large-scale numerical simulations during the last decades has not abated the fundamental role that theoretical analyses have played in understanding shock/turbulence interactions by providing closed-form solutions. In problems dealing with shock waves propagating in turbulent free streams, as in the problem treated in the present study, the most successful theoretical approach has been the linear interaction analysis (LIA) pioneered by Ribner.56–58 

Under the assumption that turbulence is comprised of small linear fluctuations that can be separated using Kovaznay's decomposition into vortical, entropic, and acoustic modes,59 LIA describes their two-way coupled interaction with the shock by using linearized RankineHugoniot jump conditions coupled with the linearized Euler equations in the post-shock gas. The resulting formalism describes the wrinkles induced by turbulence on the shock and the corresponding Kovaznay's compressible turbulence modes radiated by the interaction toward the downstream gas.

Despite its simplicity and limitations, LIA has not only provided a valuable insight into the underlying physical processes of shock/turbulence interactions, but has also worked sufficiently well for predicting the amplification of the turbulent kinetic energy (TKE), that is, commonly used for bench-marking numerical simulations.38–40 However, there exist known discrepancies between LIA and numerical simulations in the way that TKE is distributed among the diagonal components of the Reynolds stress tensor. For instance, LIA yields a smaller (larger) amplification of TKE associated with streamwise (transverse) velocity fluctuations relative to that observed in numerical simulations. These discrepancies are typically attributed to the fact that LIA treats the shock as a discontinuity, in that DNS results are observed to converge to those obtained by LIA when the ratio of the numerical shock thickness to the Kolmogorov length scale becomes sufficiently small.41,43,45

In this study, an extension that incorporates thermochemical effects of vibrational excitation and gas dissociation is made to the standard LIA previously applied to calorically perfect gases.56–58,60 As in the standard LIA, the following conditions must be satisfied: (a) the root mean square (rms) of the velocity fluctuations u needs to be much smaller than the speed of sound in both pre-shock and post-shock gases; (b) the amplitude of the streamwise displacement of the distorted shock from its mean position ξs needs to be much smaller than the upstream integral size of the turbulence ; and (c) the eddy turnover time /u needs to be much smaller than the molecular diffusion time 2/ν based on the kinematic viscosity ν, or equivalently, the turbulent Reynolds number Re=u/ν needs to be large.

In addition to the conditions [(a)–(c)] stated above, the incorporation of thermochemical effects requires that the characteristic size of the shock wrinkles, which is of the same order as , needs to be much larger than the thickness T of the thermochemical nonequilibrium region behind the shock, as depicted in Fig. 1. For instance, the value of T behind a Mach-14 normal shock at a pressure equivalent to 45 km of altitude is approximately 1 cm (see page 503 in Ref. 61). In this thermochemical nonequilibrium region, the gas relaxes toward vibrational and chemical equilibrium in an intertwined manner, in that the vibrational energy of the molecules and their dissociation probability are coupled.16,62 The value of T is approximately given by the mean post-shock velocity multiplied by the sum of the characteristic time scales of dissociation and vibrational relaxation. Since both of these characteristic time scales depend inversely on pressure and exponentially on the inverse of the temperature, the veracity of the approximation T/1 in practical hypersonic systems is expected to improve as the flight Mach number increases and the altitude decreases.

FIG. 1.

Sketch of the model problem: a normal shock wave interacts with a hypersonic free stream of weak isotropic turbulence (velocities are shown in the shock reference frame).

FIG. 1.

Sketch of the model problem: a normal shock wave interacts with a hypersonic free stream of weak isotropic turbulence (velocities are shown in the shock reference frame).

Close modal

The LIA results provided in this study yield integral formulas for the amplification of the enstrophy, composition variance, and TKE as a function of the post-shock Mach number, the density ratio, and the normalized inverse of the slope of the Hugoniot curve. The latter undergoes a change in sign at high Mach numbers due to the thermochemical effects. As a result, at Mach numbers larger than approximately 13 in the conditions tested here, a local decrement (increment) in post-shock pressure—due, for instance, to shock wrinkling—engenders an increment (decrement) in post-shock density. This peculiar structure of the Hugoniot curve at hypersonic Mach numbers is found to strongly amplify turbulence in the post-shock gas, where most of the TKE is observed to be contained in transverse velocity fluctuations of the vortical mode. For instance, the present LIA results in a maximum TKE amplification factor of approximately 2.9, whereas this value drops to 1.7 when the gas is assumed to be thermochemically frozen (i.e., diatomic calorically perfect).

The remainder of this paper is structured as follows. The RankineHugoniot jump conditions across the shock are derived in Sec. II accounting for dissociation and vibrational excitation in the post-shock gas. A linearized formulation of the problem is presented in Sec. III for the interaction of a normal shock with monochromatic vorticity disturbances. A Fourier analysis is carried out in Sec. IV to address the interaction of a normal shock with weak isotropic turbulence composed of multiple and linearly superposed vorticity modes. Finally, conclusions are given in Sec. V.

We consider first the problem of an undisturbed, normal shock wave in a cold, inviscid, irrotational, single-component gas consisting of symmetric diatomic molecules. The pre-shock density, pressure, temperature, specific internal energy, and flow velocity in the reference frame of the shock are denoted, respectively, as ρ1, P1, T1, e1, and u1. The corresponding flow variables in the post-shock gas are denoted as ρ2, P2, T2, e2, and u2.

In the reference frame attached to the shock front, the conservation equations of mass, momentum, and enthalpy across the shock are

(1a)
(1b)
(1c)

respectively. In this formulation, the symbol qd denotes a positive quantity that represents the net change of specific chemical enthalpy caused by the gas dissociation reaction

(2)

with A2 being a generic molecular species and A its dissociated atomic counterpart. In particular, qd can be expressed as

(3)

where Rg,A2 is the gas constant based on the molecular weight of A2, and Θd is the characteristic dissociation temperature. In addition, the variable α is the degree of dissociation defined as the ratio of the mass of dissociated A atoms to the total mass of the gas, or, equivalently, the mass fraction of A atoms.

Equations (1a)–(1c) are supplemented with the ideal-gas equations of state in the pre-shock gas

(4)

and in the post-shock gas

(5)

In addition, the specific internal energy in the pre-shock gas e1 is given by the translational and rotational components

(6)

whereas in the post-shock gas e2 requires consideration of translational, rotational, and vibrational degrees of freedom along with mixing between molecular and atomic species, which gives

(7)

where Θv is the characteristic vibrational temperature. The first term inside the square brackets in (7), proportional to the dissociation degree α, corresponds to the translational contribution of the monatomic species. The second term, proportional to the factor 1α, includes the translational, rotational, and vibrational contributions of the molecular species, where it has been assumed that the rotational degrees of freedom are fully activated and the molecules vibrate as harmonic oscillators.

The formulation is closed with the chemical-equilibrium condition downstream of the shock, namely,63 

(8)

where Θr is the characteristic rotational temperature, m is the atomic mass of A, kB is the Boltzmann's constant, is the reduced Planck's constant, and G=(Qela)2/Qelaa is a ratio of electronic partition functions of A atoms (Qela) and A2 molecules (Qelaa). Upon neglecting the variations of the specific internal energy with temperature due to electronic excitation, the electronic partition functions in G can be approximated as the ground-state degeneracy factors. Typical values of Θr, Θv, Θd, G, and m are provided in Table I for a wide range of molecular gases.

TABLE I.

Rotational (Θr), vibrational (Θv), and dissociation (Θd) characteristic temperatures, along with the factor G and the atomic mass m of relevant molecular gases.

H2O2N2F2I2Cl2
Θr (K) 87.53 2.08 2.87 1.27 0.0538 0.0346 
Θv (K) 6338 2270 3390 1320 308 805 
Θd (K) 51 973 59 500 113 000 18 633 17 897 28 770 
G 22/1 52/3 42/1 42/1 42/1 42/1 
m (kg) ×1026 0.167 35 2.6567 2.3259 3.1548 21.072 5.8871 
H2O2N2F2I2Cl2
Θr (K) 87.53 2.08 2.87 1.27 0.0538 0.0346 
Θv (K) 6338 2270 3390 1320 308 805 
Θd (K) 51 973 59 500 113 000 18 633 17 897 28 770 
G 22/1 52/3 42/1 42/1 42/1 42/1 
m (kg) ×1026 0.167 35 2.6567 2.3259 3.1548 21.072 5.8871 

A dimensionless formulation of the problem can be written by introducing the dimensionless parameters

(9)

along with the pressure, temperature, and density jumps

(10)

across the shock. In the expressions below, the solution for a vibrationally and chemically frozen gas (i.e., a calorically perfect diatomic gas) is recovered by taking the limits βv and βd (or α0).

Using these definitions, the dimensionless Rayleigh line

(11)

which relates P and R, is obtained by combining the mass and momentum conservation equations (1a) and (1b). In (11), the symbol M1 denotes the pre-shock Mach number defined as

(12)

where c1=(7/5)Rg,A2T1 is the speed of sound of the pre-shock gas. Regardless of the value of M1, the Rayleigh line always emanates from the pre-shock state, P=1 and R=1, as a straight line with negative slope in the {R1,P} plane.

In contrast, since the post-shock gas is calorically imperfect, its Mach number

(13)

requires a more elaborate calculation of the speed of sound

(14)

Upon substituting (5) and (7) into (14), the expression

(15)

is obtained, where

(16)

is the dimensionless component of the specific internal energy corresponding to vibrational excitation in equilibrium. In addition, the coefficients αR and αT in (15) are given by

(17)
(18)

Equation (15), along with definitions (16)–(18), determines the post-shock Mach number (13).

The equations of state (4) and (5) can be combined into a single equation as

(19)

Upon substituting (4)–(7) into the conservation equations (1a)–(1c) and using the normalizations (9) and (10), the relation

(20)

is obtained between α, R, and T. Finally, the problem is closed by rewriting the chemical-equilibrium condition (8) in dimensionless form using (9) and (10) as

(21)

which provides an additional relation between α, R, and T. In particular, given the dimensionless parameters βv, βd, and B, the combination of (19)–(21) provides the Hugoniot curve P=P(R1), which in the present case is a laborious implicit function, that is, evaluated numerically and is shown in Fig. 2. As a result, given a pre-shock Mach number M1, the post-shock state is completely determined by the intersection of the Hugoniot curve and the Rayleigh line (11).

FIG. 2.

Hugoniot curves for different molecular gases at pre-shock temperature T1=300 K and pressure P1=1 atm [gray lines: present formulation; symbols: numerical results obtained with NASA's Chemical Equilibrium with Applications (CEA) code64 excluding ionization], along with the Hugoniot curve of a gas with B=106,βv=10, and βd=100 (line colored by the degree of dissociation). The latter is compared in the inset with the Hugoniot curves of a calorically perfect monatomic gas (gray line corresponding to γ=5/3) and a calorically perfect diatomic gas (gray line corresponding to γ=7/5).

FIG. 2.

Hugoniot curves for different molecular gases at pre-shock temperature T1=300 K and pressure P1=1 atm [gray lines: present formulation; symbols: numerical results obtained with NASA's Chemical Equilibrium with Applications (CEA) code64 excluding ionization], along with the Hugoniot curve of a gas with B=106,βv=10, and βd=100 (line colored by the degree of dissociation). The latter is compared in the inset with the Hugoniot curves of a calorically perfect monatomic gas (gray line corresponding to γ=5/3) and a calorically perfect diatomic gas (gray line corresponding to γ=7/5).

Close modal

It is worth discussing some peculiarities of the Hugoniot curve, that is, obtained by including dissociation and vibrational excitation in the post-shock gas, since they are of some relevance for the shock/turbulence interaction problem studied in Secs. III and IV.

The main panel in Fig. 2 shows Hugoniot curves in light colors for H2, O2, N2, and F2 using the simple theory provided above particularized for the parameters B, βv, and βd listed in Table II. As shown in Fig. 2, the curves for O2 and N2 compare well with the more complex numerical calculations obtained with NASA's chemical equilibrium with applications (CEA) code.64 The latter incorporates variations of the specific heat with temperature due to both vibrational and electronic excitation through the NASA polynomials.65 

TABLE II.

Dimensionless parameters B, βv, and βd for relevant molecular gases at pre-shock temperature T1=300 K and pressure P1=1 atm.

H2O2N2F2I2Cl2
B×106 2.0668 6.472 14.0452 9.818 7.1796 0.6818 
βv×101 2.1127 0.7567 1.13 0.44 0.1027 0.2683 
βd×102 1.7324 1.9833 3.7667 0.6211 0.5966 0.959 
H2O2N2F2I2Cl2
B×106 2.0668 6.472 14.0452 9.818 7.1796 0.6818 
βv×101 2.1127 0.7567 1.13 0.44 0.1027 0.2683 
βd×102 1.7324 1.9833 3.7667 0.6211 0.5966 0.959 

To narrow down the exposition, the main panel in Fig. 2 also shows a Hugoniot curve colored by the degree of dissociation and obtained using the representative values B=106,βv=10, and βd=100. This is a particular choice of values that nonetheless approximately captures the order of magnitude of these parameters observed among the different gases listed in Table II (with exception of the much larger value of B observed for N2, which translates into much higher dimensionless post-shock temperatures being required to attain significant dissociation of N2).

The inset in Fig. 2 shows that the Hugoniot curve starts departing significantly from that of a calorically perfect diatomic gas [corresponding to an adiabatic coefficient γ=7/5 and a maximum density ratio R=(γ+1)/(γ1)=6] at a rather modest degree of dissociation α1% attained at M15. Despite the smallness of this crossover value of α, large changes in chemical enthalpy occur because of the large bond-dissociation specific energy of most relevant species (e.g., approximately 15 MJ/kg for O2). As a result, α1% renders αβd=O(1) in (20), which represents a balance between the heat absorbed by dissociation qd and the pre-shock internal energy e1 in the conservation equation (1c). As α is further increased, qd becomes of the same order as e2, and the departure from calorically perfect behavior becomes increasingly more pronounced.

As α becomes increasingly closer to unity, which requires the kinetic energy of the pre-shock gas to be increasingly larger than qd (or equivalently, it requires the pre-shock Mach number M1 to be increasingly larger than βd), the slope of the Hugoniot curve undergoes a change in sign and turns inward toward larger specific volumes. For the parameters investigated in Fig. 2, the turning point occurs at α0.7, where T9 (corresponding to 2700 K when T1=300 K), M113, and R12, the latter being almost double (triple) the density ratio of a calorically perfect diatomic (mono-atomic) gas. There, the inverse of the slope of the Hugoniot curve normalized with the slope of the Rayleigh line

(22)

attains a zero value. The role of Γ in the description of the shock/turbulence interaction problem will be addressed in Secs. III and IV.

As shown in Fig. 3, the value of Γ becomes negative along the upper branch of the Hugoniot curve beyond the turning point Γ = 0. Along that branch, an increment (decrement) in post-shock pressure induces a decrement (increment) in post-shock density. For the parameters tested here, the value Γ in the upper branch of the Hugoniot curve is always larger than the critical values for the onset of (a) shock instabilities associated with multi-wave66,67 and multi-valued68 solutions, and (b) D'yakov–Kontorovich pseudo-instabilities associated with the spontaneous emission of sound.8,69 Similar characteristics of the Hugoniot curve have been observed elsewhere for shocks subjected to endothermicity.70–73 

FIG. 3.

Normalized inverse of the slope of the Hugoniot curve Γ as a function of the temperature jump across the shock T for B=106,βv=10, and βd=100 (line colored by the degree of dissociation). Dashed lines represent asymptotic limits for a calorically perfect diatomic gas (βv and α0), and for a highly dissociated gas (α1).

FIG. 3.

Normalized inverse of the slope of the Hugoniot curve Γ as a function of the temperature jump across the shock T for B=106,βv=10, and βd=100 (line colored by the degree of dissociation). Dashed lines represent asymptotic limits for a calorically perfect diatomic gas (βv and α0), and for a highly dissociated gas (α1).

Close modal

Typical distributions of the density ratio R, the post-shock Mach number M2, and the pre-shock Mach number M1 are provided in Fig. 4 as a function of the temperature ratio T. The curves also show the limit behavior for α0 and βv (corresponding to a calorically perfect diatomic gas at low temperatures), and for α1 (corresponding to a fully dissociated gas at high temperatures). Some insight into these limits is provided below.

FIG. 4.

Distributions of (a) density jump R, (b) post-shock Mach number M2, and (c) pre-shock Mach number M1 as a function of the temperature jump T for B=106,βv=10, and βd=100 (lines colored by the degree of dissociation; refer to Fig. 3 for a colorbar). Dashed lines represent asymptotic limits for a calorically perfect diatomic gas (βv and α0), and for a highly dissociated gas (α1).

FIG. 4.

Distributions of (a) density jump R, (b) post-shock Mach number M2, and (c) pre-shock Mach number M1 as a function of the temperature jump T for B=106,βv=10, and βd=100 (lines colored by the degree of dissociation; refer to Fig. 3 for a colorbar). Dashed lines represent asymptotic limits for a calorically perfect diatomic gas (βv and α0), and for a highly dissociated gas (α1).

Close modal

In Fig. 4(a), the low-temperature limit of the density ratio corresponds to the standard RankineHugoniot jump condition for a calorically perfect diatomic gas

(23)

which can be derived by taking the limits α0 and βv in (20). In this low-temperature limit, the normalized slope of the Hugoniot curve becomes ΓM12, as indicated in Fig. 3.

In the opposite limit, when the post-shock gas is hot and almost fully dissociated, α1, the density jump and the normalized slope of the Hugoniot curve become

(24)

and

(25)

respectively, with βd>23/8 in the conditions tested here. At very high Mach numbers M1βd, when βd/T1, Eq. (24) simplifies to R4 in the first approximation, whereas (25) yields very small and negative values of Γ. Remarkably, unlike R,M1, and M2, the normalized inverse of the slope Γ is not bounded by its asymptotic limits at low and high Mach numbers. The relevance of this property for the problem of shock/turbulence interaction will be discussed in Secs. III and IV.

The results mentioned above for α1 indicate that the post-shock gas increasingly resembles a monatomic calorically perfect gas (corresponding to an adiabatic coefficient γ=5/3) at infinite Mach numbers, an effect that can also be visualized in Fig. 2 as the Hugoniot curve asymptotes the abscissa R11/4. However, this limit is of little practical relevance because it would require such exceedingly high temperatures that additional effects like electronic excitation, radiation, and ionization would have to be included in the formulation, thereby invalidating these considerations.

For small-amplitude velocity fluctuations and vanishing turbulent Mach numbers, the free-stream turbulence in the pre-shock gas can be represented as a linear superposition of Kovaznay's three-dimensional vorticity modes, which are solutions of the incompressible Euler equations.59,74 This section addresses the interaction of the shock with a single one of those vorticity modes.

Three reference frames are used in the analysis. Whereas the spanwise and transverse axes of all the frames coincide, the streamwise axis differs depending on whether the frames are attached to the laboratory (x), the mean shock front (xs), or the mean absolute post-shock gas motion (xc).

In the laboratory reference frame, the streamwise coordinate is denoted by x and is attached to the bulk of the pre-shock gas, which is at rest on average. In contrast, in the shock reference frame, which corresponds to the one visualized in Fig. 1, the streamwise coordinate xs moves at the mean shock velocity u1 and is therefore defined by the relation xs=xu1t in terms of the time coordinate t. The integral formulation of the conservation equations across the shock can be readily written in the shock reference frame, as done in Sec. II. Whereas the incident vorticity wave remains stationary in space in the laboratory frame, it becomes a wave traveling at velocity u1 toward the shock in the shock reference frame.

In the reference frame moving with the post-shock gas, the streamwise coordinate xc moves with the post-shock mean absolute velocity u1u2 and is therefore defined as xc=x(u1u2)t. In this frame, the vorticity and entropy fluctuations in the post-shock gas are stationary in space, which facilitates the description of the problem, as shown below.

Anticipating that the pre-shock turbulence is isotropic, there is no privileged direction of the wavenumber vector k, and therefore, the amplitude of the vorticity modes depends exclusively on k=|k|. Similarly, because of this isotropy, there is no preferred wavenumber-vector orientation relative to the shock surface. In principle, this would require the formulation of a three-dimensional problem to describe the interaction. However, a simple rotation of the reference frame can transform the problem into a two-dimensional one, as described below (see also Refs. 36, 60, and 75).

For an incident wavenumber vector arbitrary oriented in space at latitude and longitude angles θ and φ, respectively, the reference frames described in Sec. III A can be rotated counterclockwise around x by an angle equal to the longitudinal inclination of the incident wave ψ, as indicated in Fig. 5. In this way, the interaction problem becomes two-dimensional, in that all variations with respect to z are zero.

FIG. 5.

Simplification of a three-dimensional problem of a shock interacting with an arbitrary-oriented vorticity wave to a two-dimensional problem by rotating the reference frame around the streamwise axis.

FIG. 5.

Simplification of a three-dimensional problem of a shock interacting with an arbitrary-oriented vorticity wave to a two-dimensional problem by rotating the reference frame around the streamwise axis.

Close modal

Using the aforementioned rotation, the wavenumber-vector components in the streamwise and transverse directions are

(26)

respectively, with kz = 0 by construction. Similarly, in the laboratory reference frame, the vorticity vector of the incident wave in the pre-shock gas can be expressed as

(27)

with

(28)

being the vorticity amplitude in each direction. In this formulation, c2 denotes the mean speed of sound in the post-shock gas, and ε is a dimensionless velocity fluctuation amplitude, which is small in the linear theory, ε1. The vorticity of the incident wave engenders a fluctuation velocity field in the pre-shock gas given by

(29)

whose amplitude is

(30)

in the x, y, and z directions, respectively. Specifically, the z-component of the fluctuation velocity vector is uniform along z. This component will not be carried any further in the analysis, since it is transmitted unaltered through the shock because of the conservation of tangential momentum. Note also that (27) and (29) are related by the definition of the vorticity ω1=k×v1. Furthermore, the velocity field (29)–(30) is one that satisfies the incompressibility relation k·v1=0. Finally, implicit in the definitions given above is that the incident vorticity wave is inviscid, or equivalently, that the pre-shock Reynolds number of the fluctuation, 2π|v1|/(kν1), is infinitely large.

To illustrate the analysis, a particular form of the pre-shock vorticity fluctuation corresponding to the inviscid TaylorGreen vortex

(31)

is employed in the numerical results highlighted below, with ωx,1=ωy,1=0. The corresponding streamwise and transverse components of the velocity fluctuations in the pre-shock gas are given by

(32a)
(32b)

respectively. In this formulation, εu is the amplitude of the pre-shock streamwise velocity fluctuations

(33)

with ϵu1 in the linear theory.

In this linear theory, the vorticity and the streamwise and transverse velocity components in the post-shock gas reference frame are expanded to first order in εu as

(34)

respectively, with ω¯,u¯, and v¯ being the corresponding dimensionless fluctuations. The post-shock pressure and density can be similarly expressed as

(35)

with p¯ and ρ¯ being the dimensionless fluctuations of pressure and density, respectively. The brackets indicate time-averaged quantities, which are given by the solution obtained in Sec. II. In this way, all fluctuations are defined to have a zero time average.

Assuming that the Reynolds number of the post-shock fluctuations is infinitely large, the expansions (34) and (35) can be employed in writing the linearized Euler conservation equations of mass, streamwise momentum, transverse momentum, and energy as

(36a)
(36b)
(36c)
(36d)

in the reference frame moving with the post-shock gas. In this notation, the space and time coordinates have been non-dimensionalized as

(37)

The linearized Euler equations (36) can be combined into a single, two-dimensional periodically symmetric wave equation

(38)

for the post-shock pressure fluctuations. Equation (38) is integrated for τ0 within the spatiotemporal domain bounded by the leading reflected sonic wave traveling upstream, x¯c=τ, and the shock front moving downstream x¯c=M2τ, with M2=u2/c2.

In the integration of (38), the boundary condition far downstream of the shock is provided by the isolated-shock assumption, whereby the effect of the acoustic waves reaching the shock front from behind is neglected. The boundary condition at the shock front is obtained from the linearized RankineHugoniot jump conditions assuming that (a) the thickness of the thermochemical non-equilibrium region T is much smaller than the inverse of the transverse wavenumber ky1; and (b) the displacement of the shock ξs=ξs(y,t) from its mean, flat shape (see Fig. 1) is much smaller than ky1. In these limits, at any transverse coordinate y¯, the RayleighHugoniot jump conditions can be applied at the mean shock front location x¯c=M2τ and can be linearized about the mean thermochemical-equilibrium post-shock gas state P,R,T,M2, and α calculated in Sec. II, thereby yielding

(39a)
(39b)
(39c)
(39d)

In (39), ξ¯s=kyξs/εu is the dimensionless shock displacement, whereas p¯s,ρ¯s,u¯s, and v¯s are, respectively, the dimensionless fluctuations of pressure, density, streamwise velocity, and transverse velocity immediately downstream of the shock front, where thermochemical equilibrium is reached in the limit kyT1. In these relations, u¯1=u1/(εuc2) and v¯1=v1/(εuc2) are the normalized components of the pre-shock velocity field (29) engendered by the incident wave described in Sec. III B. Note that, at the turning point of the Hugoniot curve (Γ = 0), the compression of the gas exerted by the shock is isochoric in the near field and therefore leads to vanishing density fluctuations immediately downstream of the shock, as prescribed by the linearized jump condition (39d).

The flow is periodic in the transverse direction y¯. As a result, the terms involving partial derivatives with respect to y¯ in (36a), (36c), (38), and (39c) can be easily calculated from the transverse functional form of the post-shock flow variables given the incident vorticity wave (31). In particular, it can be shown that the fluctuations p¯,u¯, and ξ¯s are proportional to cos(y¯), whereas v¯ is proportional to sin(y¯). These prefactors are henceforth omitted in the analysis, but should be brought back when reconstructing the full solution from the dimensionless fluctuations.

The initial conditions required to solve (38) assume that the shock is initially flat, ξ¯s=v¯s=0 at τ = 0. Correspondingly, the initial values of the fluctuations of pressure and streamwise velocity immediately downstream of the shock must satisfy the relation u¯s+p¯s=0 at τ = 0, as prescribed by the first acoustic wave traveling upstream xc=τ. This gives a pressure fluctuation p¯s=2M2/(1+Γ+2M2) immediately downstream of the shock front at τ = 0.

The linearized problem (38), along with its boundary and initial conditions provided above, describe the fluctuations in the post-shock gas in the LIA framework. Remarkably, this problem can be integrated using the mean post-shock flow obtained from the analytical formulation provided in Sec. II, as done in the remainder of this paper, or by considering a mean post-shock flow obtained numerically with more sophisticated thermochemistry. For instance, instead of the formulation presented in Sec. II, a one-dimensional chemical equilibrium code like CEA (see Fig. 2 and Sec. II C) could be used to calculate numerically the mean post-shock conditions incorporating (a) different models for the variations of the specific heats such as the NASA polynomials,65 which include both vibrational and electronic excitation, and (b) additional chemical effects such as ionization. This can be understood by noticing that (38), along with its boundary and initial conditions, depend only on the following dimensionless parameters: the mean density jump R, the mean post-shock Mach number M2, and the inverse of the slope of the Hugoniot curve Γ, all of which can be computed numerically solving a one-dimensional shock wave subject to arbitrary thermochemistry.

At long times t(kyc2)1, the solution to the wave equation (38), subject to the boundary conditions described in Sec. III C, yields the pressure fluctuations

(40)

behind the shock. In this formulation, ω=ζ1M22 is the dimensionless frequency, where ζ is a frequency parameter defined as

(41)

with kx/ky=1/|tanθ|. Cases with ζ1 correspond to sufficiently small streamwise wavenumbers, kxky(1M22)/(M2R), whereas the opposite (sufficiently large streamwise wavenumbers) holds for ζ1. The corresponding amplitudes of the pressure wave (40) are

(42a)
(42b)
(42c)

where σb and σc are auxiliary factors defined as

(43)

To describe the far-field post-shock gas, it is convenient to split the fluctuations of velocity, pressure, and density into their acoustic (a), vortical (r), and entropic (e) components as

(44a)
(44b)
(44c)
(44d)

The acoustic pressure wave emerging from (38) is of the form p¯ae±i(ωaτκax¯y¯), where ωa and κa are the dimensionless acoustic frequency and longitudinal wavenumber reduced with c2ky and ky, respectively, which are related as

(45)

In the shock reference frame x¯=M2τ, the oscillation frequency at shock front, ω, is related to the post-shock Mach number as ω=ωaM2κa. Upon substituting this relation into (45), the expressions

(46a)
(46b)

are obtained. In (46), the solution corresponding to the positive sign in front of the square root must be excluded since it represents nonphysical acoustic waves whose amplitude increases exponentially with distance downstream of the shock when ω<(1M22)1/2.

Different forms of the solution arise depending on the value of the dimensionless frequency ω. At frequencies ω<(1M22)1/2, or equivalently ζ<1, the amplitude of the acoustic pressure decreases exponentially with distance downstream of the shock. On the other hand, for ω>(1M22)1/2, or ζ>1, the acoustic pressure becomes a constant-amplitude wave

(47)

which corresponds to a downstream-traveling sound wave for κa<0 (or ω<1), and to an upstream-traveling sound wave for κa>0 (ω>1), both cases being referenced to the post-shock gas reference frame. In this case, the acoustic modes of the density, temperature, and velocities are

(48a)
(48b)
(48c)
(48d)

respectively, where T¯=(TT2)/(εuT2) is the dimensionless post-shock temperature fluctuation.

The amplitudes of the acoustic modes of the streamwise and transverse velocity fluctuations in (48) are proportional to the amplitude of the acoustic pressure, Ua/Πs=κa/ωa and Va/Πs=1/ωa, as prescribed by second and third equations in (36). Similarly, the amplitude of the acoustic mode of the post-shock temperature fluctuations can be expressed relative to Πs as

(49)

with αR, αT, and e¯vib being defined in (17) and (18), and (16), respectively. Note that (49) simplifies to Θa/Πsγ1 in both the calorically perfect diatomic gas limit (α0 and βv, for which γ7/5) and in the fully dissociated gas limit (α1, for which γ5/3).

The entropic mode of the density fluctuations is determined by the linearized Rankine–Hugoniot jump condition (39d) after subtracting the acoustic mode

(50)

to give

(51)

in the asymptotic far field. In (51), κe=Rkx/ky is a dimensionless wavenumber, and Δj=(ΓM221)Πj is a fluctuation amplitude that depends on ζ through the pressure amplitudes Πl1,Πl2, and Πs defined in (42). Since the pre-shock gas contains only vortical velocity fluctuations, all entropic modes are generated at the shock. The entropic density fluctuations ρ¯e are related to the entropic temperature fluctuations

(52)

and both ρ¯e and T¯e induce entropic fluctuations in the degree of dissociation, as shown in (8). As a result, the thermochemical equilibrium state in the post-shock gas fluctuates depending on the local shock curvature. Specifically, there exist fluctuations of the concentrations of the chemical species A and A2 in the post-shock gas that are in phase with the entropic modes of density and temperature fluctuations. The normalized fluctuation of the degree of dissociation is

(53)

In a similar manner, the vorticity fluctuations ω¯ defined in (34) can be expressed in terms of ζ as

(54)

where, as found in the entropic perturbation field, the dimensionless rotational wavenumber is simply given by the compressed upstream wavenumber ratio κr=κe=Rkx/ky. The amplitudes are

(55a)
(55b)

where Ω1=R(1+kx2/ky2) quantifies the amplification of the preshock vorticity as a direct result of the shock compression, and Ω2=(R1)(1Γ)/(2M2) measures the vorticity production by the discontinuity front rippling. The corresponding phase for ζ<1 is given by tanϕr=Ω2Πl2/(Ω1+Ω2Πl1), which is different to that associated with entropic fluctuations tanϕe=Πl2/Πl1.

Figure 6 shows the value of |Ω|2 as a function of the shock strength M1 for six arbitrary values of the frequency parameter ζ. Three of them pertain to the long-wavelength regime ζ<1 (Ω=Ωl) and the other three to the short-wavelength regime ζ>1 (Ω=Ωs). It is found that the shape of the curve qualitatively changes depending on the wavelength regime. For instance, when compared to interactions with frequency ζ<1, cases for ζ>1 render curves with wider peaks and whose location corresponds to lower Mach numbers.

FIG. 6.

Square of the vorticity amplitude |Ω|2 as a function of the pre-shock Mach number M1 for B=106,βv=10,βd=100 and six different values of the frequency parameter: ζ=0.6, 0.7, 0.8, 1.1, 1.5, and 2.

FIG. 6.

Square of the vorticity amplitude |Ω|2 as a function of the pre-shock Mach number M1 for B=106,βv=10,βd=100 and six different values of the frequency parameter: ζ=0.6, 0.7, 0.8, 1.1, 1.5, and 2.

Close modal

The streamwise and transverse components of the vortical mode of the velocity read

(56a)
(56b)

where the phase angle is ϕr=0 for ζ>1. The amplitudes are proportional to the vorticity fluctuations as

(57a)
(57b)

where Ω depends on frequency, as shown in (55) and Fig. 6.

The weak isotropic turbulence in the pre-shock gas can be represented by a linear superposition of incident vorticity waves whose amplitudes ε vary with the wavenumber in accord with an isotropic energy spectrum E(k)=ε2(k). The root mean square (rms) of the velocity and vorticity fluctuations in the pre-shock gas can be calculated by invoking the isotropy assumption, which states that the probability the incident wave has of having orientation angles ranging from θ to θ+dθ, and from φ to φ+dφ, is proportional to the solid angle sinθdθdφ/(4π). This assumption provides the expressions

(58)

for the pre-shock rms velocity fluctuations, and

(59)

for the pre-shock vorticity fluctuations. In this section, a linear analysis is performed to calculate the variations of the rms of the velocity and vorticity fluctuations across the shock.

The analysis begins by expressing pre-shock components of the velocity fluctuation modulus as

(60a)
(60b)
(60c)

where the acoustic and vortical modes of the dimensionless velocity fluctuations in the far field are given in (48) and (56). The relations between the modes of the streamwise and transverse velocity fluctuations are provided by the irrotationality condition v¯a=κau¯a for the acoustic mode, and by the solenoidal condition kyv¯r=Rkxu¯r for the vortical mode.

The TKE amplification factor across the shock wave is defined as

(61)

where the use of (58) has been made. Furthermore, K can also be decomposed linearly into acoustic and vortical modes as K=Ka+Kr, with

(62)

The entropic mode does not contain any kinetic energy, since entropy fluctuations are decoupled from velocity fluctuations in the inviscid linear limit.

In Eq. (62), P(ζ) is a probability-density distribution given by

(63)

which satisfies the normalization 0P(ζ)dζ=1. In addition, the velocity amplitudes Ua,Ur,Va, and Vr are obtained using the long-time far-field asymptotic expressions (48) and (57). The lower integration limit of Ka is ζ = 1 since the acoustic mode decays exponentially with distance downstream of the shock in the long-wave regime ζ<1. However, the integral 1/301(Πl12+Πl12)P(ζ)dζ needs to be added to Ka when evaluating the solution in the near field x¯sx¯c1.

Figure 7 shows the TKE amplification factor K, given by the sum of the acoustic and vortical contributions in (62), as a function of the pre-shock Mach number M1. Similarly to the results observed in Sec. II, the onset of vibrational excitation at M13 begins to produce small departures of K from the thermochemically frozen result corresponding to a diatomic calorically perfect gas. These departures are exacerbated as the degree of dissociation increases and become significant even at small values of α of order 1% at M15, where K significantly departs from the curve predicted in the thermochemically frozen limit corresponding to a diatomic calorically perfect gas. The latter was shown to plateau at K=1.78 for M11 in early work,37,60 whereas the present study indicates that such plateau does not exist when thermochemical effects at hypersonic Mach numbers are accounted for.

FIG. 7.

TKE amplification factor K as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (line colored by the degree of dissociation). Dashed lines correspond to limit behavior of K calculated using the asymptotic expressions (23) and (24) for small and high Mach numbers, respectively.

FIG. 7.

TKE amplification factor K as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (line colored by the degree of dissociation). Dashed lines correspond to limit behavior of K calculated using the asymptotic expressions (23) and (24) for small and high Mach numbers, respectively.

Close modal

The resulting curve of K in Fig. 7 is non-monotonic and contains two peaks in the hypersonic range of Mach numbers. This behavior cannot be guessed by a simple inspection of the post-shock density and Mach number shown in Fig. 4. Instead, the non-monotonicity of K is related to the strong dependence of the enstrophy amplification on the wavenumber. Specifically, the vortical mode of the velocity fluctuation, which is shown below to be the most energetic, is proportional to the post-shock vorticity amplitude Ω given in (55), which peaks at different pre-shock Mach numbers depending on the frequency parameter ζ, as shown in Fig. 6.

The first peak of K reaches a value of 2.1 and occurs at M16, where α5%. In contrast, the second peak at K2.9 nearly doubles the value predicted in the thermochemically frozen limit, and occurs at a much higher Mach number M119 where dissociation is almost complete. At very large Mach numbers M1>40, in the fully dissociated regime, K asymptotes to the value K1.69 predicted for monatomic calorically perfect gases. However, as discussed in Sec. II D, this limit has to be interpreted with caution because additional thermochemical effects not included here, such as ionization and electronic excitation, play an important role at those extreme Mach numbers.

Most of the TKE produced across the shock belongs to transverse velocity fluctuations of the vortical mode. To see this, we consider the decomposition of the TKE amplification factor into longitudinal (KL) and transverse (KT) components as

(64)

with

(65a)
(65b)

The contribution of the acoustic mode to KL and KT yields negligible TKE over the entire range of Mach numbers, as shown in Fig. 8(a). In contrast, the contribution of the vortical mode is significant. Whereas the longitudinal TKE of the vortical mode KLr dominates over the transverse one KTr at supersonic Mach numbers, it plunges below KTr at hypersonic Mach numbers around the turning point of the Hugoniot curve. The value of KTr peaks at M119 with KTr3.8, as observed in Fig. 8(b). This peak is responsible for the peak in K observed Fig. 7 at the same Mach number, thereby indicating that most the TKE there is stored in vortical gas motion in the transverse direction.

FIG. 8.

(a) Acoustic and (b) vortical modes of the streamwise (KL) and transverse (KT) components of the TKE amplification factor as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (lines colored by the degree of dissociation; refer to Fig. 7 for a colorbar). Dashed lines correspond to limit behavior of KL and KT calculated using the asymptotic expressions (23) and (24) for small and high Mach numbers, respectively.

FIG. 8.

(a) Acoustic and (b) vortical modes of the streamwise (KL) and transverse (KT) components of the TKE amplification factor as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (lines colored by the degree of dissociation; refer to Fig. 7 for a colorbar). Dashed lines correspond to limit behavior of KL and KT calculated using the asymptotic expressions (23) and (24) for small and high Mach numbers, respectively.

Close modal

The mechanism whereby high-temperature thermochemistry augments the TKE across the shock in this LIA framework is explained by the linearized RankineHugoniot jump condition (39c) and is schematically shown in Fig. 9. In particular, the conservation of the tangential velocity across the wrinkled shock requires

(66)

where ϑ=π/2+arctan(ξs/y) is a local shock incidence angle whose departures from π/2 are of order ϵu, since kyξs=O(ϵu) in this linear theory. The streamwise velocity fluctuations u1 and u2 have been neglected in writing (66), since their multiplication by cosβ is smaller by a factor of order ϵu relative to the other terms. Equation (66) yields the transverse post-shock velocity fluctuation

(67)

which represents the dimensional counterpart of the linearized RankineHugoniot jump condition (39c). In Eq. (67), ξs/y<0 in both configurations sketched in Fig. 9. Note that (67) holds independently of whether the gas is thermochemically frozen or equilibrated. However, the thermochemistry influences (67) by flattening the shock front (i.e., by decreasing ξs/y) while strongly decreasing the mean post-shock velocity u2=u1/R, with the latter effect prevailing over the former. As a result, v2 and its associated kinetic energy KT are larger relative to those observed in a diatomic calorically perfect gas.

FIG. 9.

Schematics of the mechanism of TKE amplification for (a) thermochemically frozen (i.e., diatomic calorically perfect) post-shock gas, and (b) thermochemically equilibrated post-shock gas, both panels simulating the same pre-shock conditions. The flow is from right to left. The magnitude of the shock displacement and velocity perturbations has been exaggerated for illustration purposes.

FIG. 9.

Schematics of the mechanism of TKE amplification for (a) thermochemically frozen (i.e., diatomic calorically perfect) post-shock gas, and (b) thermochemically equilibrated post-shock gas, both panels simulating the same pre-shock conditions. The flow is from right to left. The magnitude of the shock displacement and velocity perturbations has been exaggerated for illustration purposes.

Close modal

The TKE amplification, along with the aforementioned decrease in the mean post-shock velocity u2 caused by the thermochemical effects, also leads to a strong amplification of the turbulence intensity across the shock. Specifically, the ratio of post- to pre-shock turbulence intensities

(68)

is found to peak at the turning point of the Hugoniot curve (α0.7,T9,M113, and R12) with a value I2/I119, as shown in Fig. 10(a). This is in contrast to the maximum value I2/I18 predicted by the theory of calorically perfect gases.

FIG. 10.

Amplification of (a) turbulence intensity and (b) turbulent Reynolds number across the shock as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (lines colored by the degree of dissociation; refer to Fig. 7 for a colorbar). The dashed lines correspond to the values of I2/I1 and Re,2/Re,1 calculated assuming that the post-shock gas is thermochemically frozen (i.e., diatomic calorically perfect).

FIG. 10.

Amplification of (a) turbulence intensity and (b) turbulent Reynolds number across the shock as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (lines colored by the degree of dissociation; refer to Fig. 7 for a colorbar). The dashed lines correspond to the values of I2/I1 and Re,2/Re,1 calculated assuming that the post-shock gas is thermochemically frozen (i.e., diatomic calorically perfect).

Close modal

Although the theory presented above is formulated in the inviscid limit, the ratio of post- to pre-shock turbulent Reynolds numbers

(69)

is a finite quantity that can be calculated. In the last term of (69), the use has been made of the fact that the only wavenumber, that is, distorted through the shock is the longitudinal one, which changes from kx in the pre-shock fluctuations to kxR in the post-shock ones. In addition, the molecular viscosity is assumed to vary with temperature raised to the power of 0.7.

Remarkably, the vortical post-shock fluctuations downstream of the hypersonic shock are not only much more intense than those upstream, but they also have a higher turbulent Reynolds number Re,2>Re,1, as shown in Fig. 10(b). Similarly to the turbulence intensities, the maximum ratio of turbulent Reynolds numbers across the shock is reached at the turning point of the Hugoniot curve (α0.7,T9,M113, and R12) with a value of Re,2/Re,15. In contrast, the theory of calorically perfect gases predicts an attenuation of the turbulent Reynolds number at those conditions. When thermochemical effects are accounted for, the amplification of the turbulent Reynolds number lasts until M120, beyond which the increase in post-shock temperature and the decrease in post-shock density make Re,2/Re,1 to plummet below unity.

In summary, the increase in transverse velocity fluctuations of the vortical mode across the shock is responsible for the TKE amplification in this linear theory. In addition, the results indicate that the TKE is more amplified when dissociation and vibrational excitation are accounted for at high Mach numbers. In the conditions tested here, the post-shock fluctuations resulting at hypersonic Mach numbers can be—at most—19 times more intense and can have—at most—a five times larger turbulent Reynolds number than the pre-shock fluctuations.

The weak isotropic turbulence in the pre-shock gas becomes anisotropic as it traverses the shock wave. An anisotropy factor that quantifies this change can be defined as60 

(70)

with 1Ψ1. The cases Ψ=1 and 1 represent anisotropic turbulent flows dominated by longitudinal and transverse velocity fluctuations, respectively. In contrast, Ψ=0 corresponds to an isotropic turbulent flow, KT=KL=K. Figure 11 shows that dissociation and vibrational excitation dissociation lead to larger anisotropy factors in the post-shock gas compared to the thermochemically frozen (diatomic calorically perfect) case in the relevant range of hypersonic Mach numbers up to the fully dissociated gas limit.

FIG. 11.

Anisotropy factor Ψ as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (line colored by the degree of dissociation). Dashed lines correspond to limit behavior of Ψ calculated using the asymptotic expressions (23) and (24) for small and high Mach numbers, respectively.

FIG. 11.

Anisotropy factor Ψ as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (line colored by the degree of dissociation). Dashed lines correspond to limit behavior of Ψ calculated using the asymptotic expressions (23) and (24) for small and high Mach numbers, respectively.

Close modal

The vortical motion downstream of the shock is quantified by the enstrophy amplification factor

(71)

where the use of (59) and of the invariance of the normal vorticity across the shock has been made. In (71), W is the enstrophy amplification factor in the transverse direction

(72)

with

(73)

being the amplification factor of the rms of the zcomponent of the vorticity. Equation (73) includes the asymptotic amplitudes defined in (55) and the relation

(74)

The enstrophy amplification factor W is provided in Fig. 12 as a function of the pre-shock Mach number. Similarly to Fig. 7 for K, the curve of W displays two maxima, but the differences with respect to the thermochemically frozen case are much larger for W. The first peak of W is dominated by the increase in short-wavelength vorticity, as shown in Fig. 6, and it represents an amplification of nearly four times the enstrophy predicted by the theory of calorically perfect gases.

FIG. 12.

Enstrophy W as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (line colored by the degree of dissociation). Dashed lines correspond to limit behavior of W calculated using the asymptotic expressions (23) and (24) for small and high Mach numbers, respectively.

FIG. 12.

Enstrophy W as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (line colored by the degree of dissociation). Dashed lines correspond to limit behavior of W calculated using the asymptotic expressions (23) and (24) for small and high Mach numbers, respectively.

Close modal

Whereas the pre-shock density is uniform because of the vortical character of the incident modes, the density in the post-shock gas fluctuates due to both acoustic and entropic modes generated by the shock wrinkles. To investigate these fluctuations, we consider the normalized density variance

(75)

which depends on the integral of the energy spectrum E over the entire wavenumber space. The prefactors Ga and Ge represent density-variance components induced by acoustic and entropic modes, respectively, and are given by

(76a)
(76b)

where the use of (51) has been made. Figure 13(a) shows that, while the vortical fluctuations across the shock are increased by dissociation, the density variance induced by the entropic mode is small for M110 but increases sharply thereafter up to M119, where it achieves a maximum value. As observed by comparing Figs. 8(b) and 13(a), the acoustic prefactor Ga is found to be negligible compared to the entropic one Ge.

FIG. 13.

Entropic prefactors of (a) the post-shock density variance and (b) the post-shock degree of dissociation as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (lines colored by the degree of dissociation; refer to Fig. 12 for a colorbar). Dashed lines correspond to limit behavior of Ge calculated using the asymptotic expressions (23) and (24) for small and high Mach numbers, respectively.

FIG. 13.

Entropic prefactors of (a) the post-shock density variance and (b) the post-shock degree of dissociation as a function of the pre-shock Mach number M1 for B=106,βv=10, and βd=100 (lines colored by the degree of dissociation; refer to Fig. 12 for a colorbar). Dashed lines correspond to limit behavior of Ge calculated using the asymptotic expressions (23) and (24) for small and high Mach numbers, respectively.

Close modal

Whereas the RankineHugoniot jump condition (39d) evaluated at the turning point of the Hugoniot curve Γ = 0 indicates that the density fluctuations immediately downstream of the shock are zero, the entropic prefactor in Fig. 13(a) at M113 (where Γ vanishes) leads to a non-zero density variance. The two results can be reconciled by noticing that the formulation in (76) and the approximation GeGa are applicable only to the far-field downstream of the shock. In contrast, the acoustic mode needs to be retained near the shock. Specifically, the post-shock density fluctuations in the near field vanish as Γ0 because of a destructive interference between the acoustic and entropic modes. In contrast, the entropic mode dominates in the far field and leads to non-zero post-shock density fluctuations.

The entropic component of the density variance engenders a variance of the degree of dissociation given by

(77)

where

(78)

is the corresponding prefactor. Figure 13(b) shows that Ae attains a maximum value at M115, and becomes negligible both in the absence of dissociation and when dissociation is complete.

The interaction between a hypersonic shock wave and weak isotropic turbulence has been addressed in this work using LIA. Contrary to previous studies of shock/turbulence interactions focused on calorically perfect gases, the results provided here account for endothermic thermochemical effects of vibrational excitation and gas dissociation enabled by the high post-shock temperatures. Important approximations used in this theory are that the thickness of the thermochemical non-equilibrium region trailing the shock front is small compared to the characteristic size of the shock wrinkles, and that all fluctuations in the flow are small relative to the mean.

The results presented here indicate that the thermochemical effects act markedly on the solution in a number of important ways with respect to the results predicted by the theory of calorically perfect gases:

  • Significant departures from calorically perfect-gas behavior can be observed in the solution even at modest degrees of dissociation of 1%, corresponding to Mach 5 and therefore to the beginning of the hypersonic range. This is because the associated bond-dissociation energies of typical molecules are large. As a result, the chemical enthalpy invested in dissociation in the post-shock gas can easily surpass the pre-shock thermal energy and become of the same order as the pre-shock kinetic energy.

  • A turning point in the Hugoniot curve is observed at approximately Mach 13 and 70% degree of dissociation that leads to a significant increase of the mean post-shock density of approximately 12 times its pre-shock value, which represents nearly twice the maximum density jump predicted by the theory of calorically perfect gases.

  • The aerothermodynamic behavior of the post-shock gas changes fundamentally around the turning point in the Hugoniot curve. As the Mach number increases above 13, positive fluctuations of streamwise velocity engender positive pressure fluctuations in the post-shock gas that are accompanied by negative density fluctuations. In this way, the local post-shock density and pressure are anticorrelated, although the shock remains stable to corrugations in all operating conditions tested here.

  • The amplification of TKE is larger than that observed in calorically perfect gases. Whereas the streamwise velocity fluctuations across the shock are decreased, the transverse ones are greatly increased (i.e., much more than in a diatomic calorically perfect gas). This phenomenon can be explained in the linear theory by using the conservation of tangential momentum, which elicits larger transverse velocity fluctuations as a result of the increase in post-shock density that occurs due to dissociation and vibrational excitation. This effect also leads to a much more significant increase of anisotropy and enstrophy across the shock than that observed in a diatomic calorically perfect gas.

  • Most of the amplified content of TKE is stored in vortical velocity fluctuation modes in the post-shock gas. The trend of the TKE amplification factor with the pre-shock Mach number is non-monotonic and involves two maximum values, one equal to 2.1 at Mach 6 (corresponding to a degree of dissociation of 5%), and a second one equal to 2.9 at Mach 19 (corresponding to a degree of dissociation larger than 99%).

  • The turbulence intensity and turbulent Reynolds number increase across the shock and reach maximum amplification factors of 19 and 5, respectively, both occurring at the turning point of the Hugoniot curve (Mach 13 and degree of dissociation of 70%). The maximum amplification factor of the turbulence intensity is more than twice the one attainable in a diatomic calorically perfect gas. The amplification of the turbulent Reynolds number observed here is in contrast with the attenuation predicted by the theory of calorically perfect gases at hypersonic Mach numbers.

  • The density variance in the post-shock gas is almost exclusively generated by entropic modes radiated by the shock wrinkles and nearly doubles the value predicted for calorically perfect gases. Similarly, the shock front generates entropic fluctuations of the concentration of atomic species that might be relevant for applications in supersonic combustion if the post-shock gas is going to be employed to oxidize the fuel.14,76

The LIA predictions for the overall TKE amplification factor in calorically perfect gases have been previously found to be in fair agreement with numerical simulations36,38,39,49 and experiments.33 However, the way LIA predicts the amplified TKE to be partitioned in the streamwise and transverse directions has not been as successful. In particular, computational and experimental studies at supersonic Mach numbers have often reported Reynolds stress tensors with dominant streamwise contributions, this being an effect typically attributed to convective non-linearities and molecular transport.49 Whether these discrepancies subside or persist at hypersonic Mach numbers is an open question of research.

The theoretical results provided here indicate amplified levels of post-shock fluctuation energies that could perhaps be unexpected at first, because of the high post-shock temperatures prevailing at hypersonic Mach numbers. These findings would greatly benefit from comparisons with simulations and experiments in future studies.

This theory could be extended to include additional phenomena such as: (a) non-equilibrium vibrational relaxation and finite-rate dissociation;16,77–79 (b) multi-component gas mixtures (particularly O2 and N2 for shock/turbulence interactions in air); (c) compressibility and anisotropy in the pre-shock turbulence; (d) the effects of walls downstream of the shock to address modal resonance in high-temperature inviscid shock layers around hypersonic projectiles; and (e) electronic excitation, radiation, and ionization in the post-shock gas for hypersonic flows at orbital stagnation enthalpies.

C.H. was funded by a 2019 Leonardo Grant for Researchers and Cultural Creators awarded by the BBVA Foundation, and by the MICINN Grant No. PID2019–108592RB-C41.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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