Motivation For High-Enthalpy Transition Study

List of Tables

Chapter 1 Introduction

1.3 Motivation For High-Enthalpy Transition Study

In hypervelocity flow over cold, slender bodies, the most significant instability mech- anism is the so-called second or Mack mode. These flows are characteristic of atmo- spheric re-entry and air-breathing hypersonic vehicles, and were the target conditions for which high-enthalpy facilities like the T5 shock tunnel at Caltech were developed.

Section 4.3.2 presents a more thorough discussion of Mack’s second mode, as well as computations demonstrating that this mode is present and unstable for T5 conditions.

A second mode disturbance depends on the amplification of acoustic waves trapped in the boundary layer, as described by Mack (1984). Another potential disturbance is the first mode, which is the high speed equivalent of the viscous Tollmien–Schlichting instability (Malik, 2003). However, at high Mach number (> 4) and for cold walls, the first mode is damped and higher modes are amplified, so that the second mode would be expected to be the only mechanism of linear instability leading to transition for a slender cone at zero angle of attack.

1.3.1 Damping of Acoustic Disturbances by Vibrational Re- laxation

By assuming that the boundary layer acts as an acoustic waveguide for disturbances (see Fedorov (2011a) for a schematic illustration of this effect), the frequency of the most strongly amplified second-mode disturbances in the boundary layer may be estimated as Equation (1.1), as shown in Stetson (1992) with a different coefficient².

f ≈0.7 U_e

2δ₉₉ (1.1)

Here δ₉₉ is the boundary layer thickness defined by the height above the surface

2Stetson (1992) reports 0.8; in the present work 0.7 is found to more closely match STABL computations for most-amplified frequency.

where the local velocity is 99% of the freestream velocity, andU_eis the velocity at the boundary layer edge. For a typical T5 condition in air, with enthalpy of 10 MJ/kg and reservoir pressure of 50 MPa, the boundary layer thickness is on the order of 1.5 mm by the end of the cone and the edge velocity is 4000 m/s. This indicates that the most strongly amplified frequencies are in the 1 MHz range. This is broadly consistent with the results of Fujii and Hornung (2001).

Kinsler et al. (1982) provide a good general description of the mechanisms of attenuation of sound waves in fluids due to molecular exchanges of energy within the medium. The relevant exchange of energy for carbon dioxide in the boundary layer of a thin cone at T5-like conditions is the conversion of molecular kinetic energy (e.g., from compression due to acoustic waves) into internal vibrational energy. In real gases, molecular vibrational relaxation is a nonequilibrium process, and therefore irreversible. This absorption process has a characteristic relaxation time.

The problem of sound propagation, absorption, and dispersion in a dissociating gas has been treated from slightly different perspectives by Clarke and McChesney (1964), Zeldovich and Raizer (1967), and Kinsler et al. (1982). However, in nonequilibrium flows when the acoustic characteristic time scale and relaxation time scale are similar, some finite time is required for molecular collisions to achieve a new density under an acoustic pressure disturbance. This results in a work cycle, as the density changes lag the pressure changes. The area encompassed by the limit cycle’s trajectory is related to energy absorbed by relaxation. Energy absorbed in this way is transformed into heat and does not contribute to the growth of acoustic waves (Leyva et al., 2009b).

1.3.2 Relevant Properties of Air, N

₂

, and CO

₂

In order to damp acoustic vibrations within the boundary layer, energy must be trans- ferred into the gas molecules’ internal modes, the energy content of which depends upon vibrational specific heat. Vincenti and Kruger (1965) present Equation (1.2)

for vibrational specific heat, where Θ_i is the characteristic vibrational temperature of each mode of the gas molecule, and R is the molecule’s gas constant. The expo- nential factors dominate the vibrational contribution from each mode, and indicate that an increase in temperature causes an increase in both total specific heat and the contribution to specific heat from each vibrational mode.

C_vvib =R

i

Θ_i T

2

e^Θi/T (e^Θi/T −1)²

(1.2) Specifically, as Θ_i/T becomes large (for small T), the summand tends to zero, which means there is no contribution to the vibrational specific heat from that vibra- tional mode. As Θ_i/T becomes small (for largeT), the summand tends to unity, and the maximum contribution from a given vibrational mode is thereforeR. As temper- ature increases within the boundary layer, each mode becomes more fully excited and capable of exchanging more energy from acoustic vibrations. The temperature of the boundary layer increases with enthalpy. See Figure 5.7 in Chapter 5 for an illustra- tion of the dependence upon reservoir enthalpy ofT^∗, a characteristic boundary layer reference temperature, for each experiment.

Carbon dioxide, a linear molecule, has four normal vibrational modes. The first two, which correspond to transverse bending, are equal to each other, and have char- acteristic vibrational temperatures Θ₁ = Θ₂= 959.66 K. The third mode, correspond- ing to symmetric longitudinal stretching, has Θ₃ = 1918.7 K, and the fourth mode, corresponding to asymmetric longitudinal stretching, has Θ₄ = 3382.1 K.

Camac (1966) assumed that the four vibrational modes for carbon dioxide could be modeled as relaxing at the same rate due to inter-mode coupling, and proposed a single formula, Equation (1.3), to calculate vibrational relaxation times for all four modes, which was reproduced in Fujii and Hornung (2001).

ln (A₄τ_CO2P) =A₅T^−1/3 (1.3)

HereA₄andA₅are constants given by Camac for carbon dioxide asA₄= 4.8488×10²Pa⁻¹s⁻¹ and A₅ = 36.5 K^1/3. Using the constants suggested by Camac, withP = 35 kPa and

T = 1500 K, which are consistent with a typical T5 condition with reservoir enthalpy 10 MJ/kg and reservoir pressure 50 MPa, we find vibrational relaxation time τ_CO2

= 1.43×10⁻⁶ s, which indicates that frequencies around 700 KHz should be most strongly absorbed at these conditions. This is broadly similar to the results of Fujii and Hornung (2001), who computed absorption curves at 1000 K and 2000 K with peaks bracketing 700 kHz.

1.3.3 Gas in Chemical Nonequilibrium at Rest

The method of Fujii and Hornung (2001) for estimating the absorption of acoustic waves perturbing high temperature gas is used to compute sample absorption curves for several conditions from the present study, chosen to match the reference tempera- ture of the boundary layer and the computed most-amplified frequency at transition for each case. One condition each is computed for an air, N₂, and 50% CO₂ experi- ment. Figure 1.1 presents the results of these computations. In the relevant frequency range, the computed acoustic absorption per wavelength for 50% CO2 is more than 3 orders of magnitude larger than the air case at similar conditions, and about 5 orders of magnitude larger than the N₂ case.

Thus, in a flow of gas that absorbs energy most efficiently at frequencies similar to the most strongly amplified frequencies implied by the geometry of the boundary layer, laminar to turbulent transition is expected to be delayed. Using computational techniques described in Chapter 4, we show that the flow of carbon dioxide/air mix- tures over a slender cone at T5 conditions allows for such a match in frequencies and a significant effect on the predicted stability properties of the boundary layer. We then perform a series of experiments to confirm this effect for transition onset, the results of which are presented in Chapter 5.

Frequency [Hz]

Absorption per wavelength

10⁰ 10² 10⁴ 10⁶

10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

2776 (N2: T^∗= 1295 K,ΩTr= 1054 kHz) 2762 (Air: T^∗= 1252 K,ΩTr= 983 kHz) 2812 (CO250%: T^∗= 1236 K,ΩTr= 924 kHz)

Figure 1.1: Fujii acoustic absorption per wavelength for air, N₂, and 50% CO₂ calcu- lated at similar conditions. The area of the graph highlighted in gray, which extends from 100 kHz to 2 MHz, is the most relevant frequency range for the present study and encompasses the predicted most amplified frequencies at transition for all of the included cases. In this range, the computed acoustic absorption per wavelength for 50% CO₂ is more than 3 orders of magnitude larger than the air case.