19-Thermal non-equilibrium

Vortex roll up in mixing layer. (a) Passive scalar, (b) Tv , (c) κ for 1 ev , (d) κ for 3 ev.

Thermal non-equilibrium of a high-enthalpy free shear layer during breakdown to turbulence

Ali Musawi *, Neil D. Sandham *

m.a.musawi@soton.ac.uk , n.sandham@soton.ac.uk

* University of Southampton, Boldrewood Innovation Campus, Burgess Rd, Southampton SO16 7QF, United Kingdom

Key Words: Thermal Non-Equilibrium, Hypersonic, Turbulence, Mixing Layer, Direct Numerical Simulation



Under the high-enthalpy conditions seen in hypersonic flight, non-equilibrium effects caused by vibrational excitation and chemical reactions further complicate the problem of prediction of transition to turbulence.Relatively little is known about the extent to which the effects need to be incorporated into models for the transition process and for the resulting turbulent flow.

The objective of the present work is twofold.

  • Firstly, we assess the impact of different model choices, here focusing on thermal non-equilibrium and addressing the question of whether it is significantly better to use separate vibrational energy equations for each molecule, compared with a combined single vibration energy equation.
  • Secondly, for a simple model shear flow, we want to understand the physical origin of thermal non-equilibrium relative to the process of vortex roll-up and entrainment. This understanding can then aid in building physics-based models.

Multi-temperature simulations of flows in thermal non-equilibrium, such as the model proposed by Gnoffo et al. [1989] and that used by Passiatore et al. [2022], typically make use of a single conservation equation for the vibrational energy, effectively combining the rate of relaxation of the molecules.

While it brings benefits such as less computational expense, the extent to which this will influence the result has not been investigated in the context of high-enthalpy transition to turbulence. In the present work, a separate vibrational energy conservation equation was considered for each molecule, with relaxation times computed using the expression given by Millikan and White [1963].

Figure 1 shows a heat bath (0D) case where a frozen mixture consisting of 21% O2 and 79% N2 is relaxed to equilibrium, starting from a prescribed condition with ro-translational temperature T = 6900K and vibrational temperature Tv = 800K.

The case compares the molecular relaxation of three models: a single weighted averaged relaxation time with a single vibrational energy (ev) conservation equation (as in Gnoffo et al. [1989], shown with the red lines), a multi relaxation time used with a single ev equation (as in Passiatore et al. [2022] shown with symbols), and a model with a different equation for each molecule (solid line).

While the first two methods are almost identical, the multi equation simulation demonstrates a slower relaxation due to a larger contribution of the slower relaxation of N2 which is not correctly captured in the one-equation models.

To further study the effect of the formulation, we consider transition to turbulence in a mixing layer, initially considering the two-dimensional development of large vortical structures. The simulation evolves temporally with periodic streamwise boundary conditions, initialised with an equilibrium thermal condition and a small disturbance to initiate a vortex roll-up based on a Kelvin-Helmholtz instability.

The scale of the problem has been set so that the flow timescale is comparable to the vibrational relaxation scales, which are both much faster than the chemistry timescales, with flow conditions specified in Table 1.

Figure 2(a) plots contours of a passive scalar that illustrates the developed vortex at t = 0.8 milliseconds. As the vortex develops, local regions of the flow are shifted out of equilibrium.

Figure 2(b) plots the vibrational temperature; the temperature at the centre of the roll-up is reduced whereas the stagnation region in between successive vortices experiences an increased temperature. As the roll-up continues, the centre of the roll reaches equilibrium while the surrounding regions are driven out of equilibrium.

To investigate this in more detail, a dimensionless non-equilibrium parameter is defined as

where T is the translational temperature and Tv is the vibrational temperature.

When the flow has zero contribution from vibrational energy, κ tends to minus one and when the flow is predominately composed of vibrational energy κ tends to one. In the case in which the region is at equilibrium, κ is zero. The resulting structure is shown on figures 2(c) and (d) for the single and multi vibration equation cases.

Essentially they show the difference between the ro-translational temperature and the vibrational temperature, illustrating that the vibrational temperature of fluid decelerating towards the stagnation point and then being entrained into the vortex (the grey regions) lags the ro-translational temperature, while similar lag effects happen for fluid being accelerated around the vortex (the red regions).

It is also notable that the stagnation point itself remains in equilibrium, presumably since the local flow timescale, associated with fluid slowing down to stagnation conditions, is longer than the vibrational relaxation time. Additionally, substantial differences are seen between the two formulations where higher thermal non-equilibrium is present due to a slower relaxation of the mixture with the multi-equation formulation.

Table 1: Simulation properties for Figure 2.

Table 1: Simulation properties for Figure 2.


Relaxation to equilibrium of 0D heat bath case. Ttr=’- -’, Tv=’-’, NR: Newton-Raphson

Figure 1 :Relaxation to equilibrium of 0D heat bath case. Ttr=’- -’, Tv=’-’, NR: Newton-Raphson


Vortex roll up in mixing layer. (a) Passive scalar, (b) Tv , (c) κ for 1 ev , (d) κ for 3 ev.

Vortex roll up in mixing layer. (a) Passive scalar, (b) Tv , (c) κ for 1 ev , (d) κ for 3 ev.


To conclude, the difference between a single and multi ev conservation equation in a 0D case demonstrated the impact of modeling the vibrational relaxation of each molecule separately, capturing the dominant influence of the slower relaxation time of N2 . In the 2D shear layer, the initial departure from equilibrium starts from the core of the roll whereas the formation of thermal non-equilibrium continues in the surrounding of the flow. Additionally, the parameter κ was introduced as a universal measure of thermal non-equilibrium.

The different models assessed in the abstract will be explained in the presentation, in addition to highlighting the sources of the discrepancy. The coupling between turbulence and thermal non-equilibrium will be further demonstrated for cases with different relaxation time scales. The work is currently being extended to cover 3D breakdown to turbulence.


Gnoffo, P. A., Gupta, R. N. and Shinn, J. L. [1989], Conservation Equations and Physical Models For Hypersonic Air Flows In Thermal and Chemical Nonequilibrium, Technical Report 2867, NASA.

Millikan, R. C. and White, D. R. [1963], ‘Systematics of Vibrational Relaxation’, The Journal of Chemical Physics 39(12), 3209–3213.

Passiatore, D., Sciacovelli, L., Cinnella, P. and Pascazio, G. [2022], ‘Thermochemical Non-Equilibrium Effects In Turbulent Hypersonic Boundary Layers’, Journal of Fluid Mechanics 941, A21.