Theory of turbulence augmentation across hypersonic shock waves

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Theory of turbulence augmentation across hypersonic shock waves
74th Annual Meeting of the APS Division of Fluid Dynamics
Alberto Cuadra 1, a) , César Huete1, Marcos Vera1, & Javier Urzay2
a)acuadra@ing.uc3m.es
1Grupo de Mecánica de Fluidos, Universidad Carlos III de Madrid
2Center for Turbulence Research, Stanford University
Phoenix |November 21-23, 2021
Huete: Spanish MCINN and BBVA Foundation (Leonardo Grant)
Urzay: US AFOSR and US DoE/NNSA

Theory of turbulence
augmentation across
hypersonic shock
waves
A. Cuadra (presenter)
C. Huete, M. Vera,
J. Urzay
1Motivation
Effects of dissociation
and vibrational
excitation on the mean
post-shock quantities
LIA of turbulence
interacting with
hypersonic shocks
Conclusion
Motivation
∆p
p∼∆ρ
ρ=O(105),∆T
T=O(1).
In hypersonic flight near the ground,
the Reynolds number becomes large
because of the comparatively larger
densities
∆Re
Re =O(105).Urzay, J.,& Di Renzo, M. (2021). Annual Research Briefs, Center for Turbulence Research, 7-32.

Theory of turbulence
augmentation across
hypersonic shock
waves
A. Cuadra (presenter)
C. Huete, M. Vera,
J. Urzay
2Motivation
Effects of dissociation
and vibrational
excitation on the mean
post-shock quantities
LIA of turbulence
interacting with
hypersonic shocks
Conclusion
Motivation
Urzay,J., & Di Renzo, M. (2021). Annual Research Briefs, Center for Turbulence Research, 7-32.
Hypersonic flight at low altitudes is characterized by:
• High free-stream Mach numbers Ma ≥5•large normal Mach numbers
• High free-stream and post-shock •turbulent boundary layers
unit Reynolds numbers Re ∼107−109m-1
• High stagnation enthalpies h0∼5−30 MJ/kg •much higher than the vibrational specific energies
of O2and N2
• Small mean free paths λ∼0.1µm•short vibrational relaxation distances

Theory of turbulence
augmentation across
hypersonic shock
waves
A. Cuadra (presenter)
C. Huete, M. Vera,
J. Urzay
Motivation
3Effects of dissociation
and vibrational
excitation on the mean
post-shock quantities
LIA of turbulence
interacting with
hypersonic shocks
Conclusion
Effects of dissociation and vibrational excitation on the mean post-shock quantities
Integral conservation equations accross shock
waves in dissociating gases
ρ1u1=ρ2u2,
p1+ρ1u2
1=p2+ρ2u2
2,
e1+p1/ρ1+u2
1/2 = e2+p2/ρ2+u2
2/2 + qd,
Upstream flow is sufficiently cold to be
approximated as
e1= (5/2)Rg,A2T1, p1=ρ1Rg,A2T1.
where αis the degree of dissociation,
defined as the mass fraction of Aatoms in
the reaction A2⇌A + A,that must be
solved with the aid of the chemical
equilibrium condition.

Theory of turbulence
augmentation across
hypersonic shock
waves
A. Cuadra (presenter)
C. Huete, M. Vera,
J. Urzay
Motivation
3Effects of dissociation
and vibrational
excitation on the mean
post-shock quantities
LIA of turbulence
interacting with
hypersonic shocks
Conclusion
Effects of dissociation and vibrational excitation on the mean post-shock quantities
Integral conservation equations accross shock
waves in dissociating gases
ρ1u1=ρ2u2,
p1+ρ1u2
1=p2+ρ2u2
2,
e1+p1/ρ1+u2
1/2 = e2+p2/ρ2+u2
2/2 + qd,
p2=ρ2Rg,A2T2(1 + α), qd=αRg,A2Θd,
e2=Rg,A2T23α+ (1 −α)5
2+Θv/T2
eΘv/T2−1,
α2
1−α=GmΘrπmkB
ℏ2
3/2√T2
ρ2
e−
Θd
T21−e−
Θv
T2,
where αis the degree of dissociation,
defined as the mass fraction of Aatoms in
the reaction A2⇌A + A,that must be
solved with the aid of the chemical
equilibrium condition.

Theory of turbulence
augmentation across
hypersonic shock
waves
A. Cuadra (presenter)
C. Huete, M. Vera,
J. Urzay
Motivation
4Effects of dissociation
and vibrational
excitation on the mean
post-shock quantities
LIA of turbulence
interacting with
hypersonic shocks
Conclusion
Effects of dissociation and vibrational excitation on the mean post-shock quantities
Endothermicity due to dissociation and
vibrational excitation does the following:
•increases the mean post-shock density
•decreases the mean post-shock velocity
•decreases the mean post-shock Mach
number
•decreases the mean post-shock
temperature
Jump conditions

Theory of turbulence
augmentation across
hypersonic shock
waves
A. Cuadra (presenter)
C. Huete, M. Vera,
J. Urzay
Motivation
Effects of dissociation
and vibrational
excitation on the mean
post-shock quantities
5LIA of turbulence
interacting with
hypersonic shocks
Conclusion
LIA of turbulence interacting with hypersonic shocks
Considering:
•turbulence is comprised of small fluctuations,
•Kovasznay decomposition into vortical, entropic and acoustic modes,
we can solve this problem analytically by using
•linearized Rankine-Hugoniot relations,
•linearized Euler equations in the post-shock gas.
Limits of validity
Assumptions standard LIA:
(a) rms(uℓ)≪a1and a2,
(b) ξs≪ℓ,
(c) ℓ/uℓ≪ℓ2/ν.
With thermochemical effects:
(d) ℓT≪ℓ
For a given ℓ, this condition
becomes increasingly more
accurate as the altitude
decreases.

Theory of turbulence
augmentation across
hypersonic shock
waves
A. Cuadra (presenter)
C. Huete, M. Vera,
J. Urzay
Motivation
Effects of dissociation
and vibrational
excitation on the mean
post-shock quantities
6LIA of turbulence
interacting with
hypersonic shocks
Conclusion
LIA of turbulence interacting with hypersonic shocks: Calorically Perfect Gas
Turbulent Kinetic Energy Turbulence intensity and Turbulent
Reynolds number
At hypervelocities (Ma ≳10), the calorically perfect gas approximation predicts a saturation
in the amplification of kinetic energy and turbulence intensity, along with a decrease in the
turbulent Reynolds number accross the shock.

Theory of turbulence
augmentation across
hypersonic shock
waves
A. Cuadra (presenter)
C. Huete, M. Vera,
J. Urzay
Motivation
Effects of dissociation
and vibrational
excitation on the mean
post-shock quantities
7LIA of turbulence
interacting with
hypersonic shocks
Conclusion
LIA of turbulence interacting with hypersonic shocks: Vibrationally Excited, Dissociating Gas
Turbulent Kinetic Energy Turbulence intensity and Turbulent
Reynolds number
In contrast, the incorporation of dissociation and vibrational excitation predicts larger kinetic
energy and turbulence intensity amplification rates, along with an increase in the turbulent
Reynolds number accross the shock.

Theory of turbulence
augmentation across
hypersonic shock
waves
A. Cuadra (presenter)
C. Huete, M. Vera,
J. Urzay
Motivation
Effects of dissociation
and vibrational
excitation on the mean
post-shock quantities
8LIA of turbulence
interacting with
hypersonic shocks
Conclusion
LIA mechanism of turbulence amplification across hypersonic shocks
Calorically perfect gas Vibrationally Excited, Dissociating Gas
Conservation of tangential momentum dictates that the transverse velocity fluctuations
should increase across the shock – these are larger at hypersonic velocities because of the
associated larger post-shock densities induced by endothermic thermochemical effects.

Conclusions
Key takeaways
•Significant departures from calorically perfect gas behaviour can be observed in the
solution even at modest degrees of dissociation of 1%.
•Aturning point in the Hugoniot curve is observed at approximately Mach 13 and
70% degree of dissociation.
•The amplification of TKE doubles that observed in calorically perfect gases, with
most of the content of TKE downstream in form of vortical modes.
•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 calorically perfect case.
•Thermochemical effects arising at hypersonic velocities appear to enhance
turbulent fluctuations in the post-shock gas.
Huete, C. et al. (2021). Thermochemical effects on hypersonic shock
waves interacting with weak turbulence. Physics of Fluids, 33(8), 086111