2. Equations for a turbulent mixing layer
For the case of a thin, steady state, high Mach number, radiative mixing layer, the advective terms along the direction of the mean flow can be neglected with respect to the corresponding terms across the width of the mixing layer (see the schematic diagram in Fig. 1). This is a result of the fact that the opening angle of the mixing layer is much smaller than the Mach angle of the mixing layer (see Cantó & Raga 1991). Under this approximation, the momentum and energy equations can be written as:
where y is a coordinate measured from the jet beam into the mixing layer (see Fig. 1), v is the mean velocity (directed parallel to the jet beam), L is the radiative energy loss per unit volume, and µ and are the turbulent viscosity and conductivity, respectively (which are assumed to be constant throughout the cross-section of the mixing layer).
Eq. (1) can be integrated to obtain the linear Couette flow solution
where is the jet velocity, and h is the local width of the mixing layer (see Fig. 1). This solution can be substituted in Eq. (2), which can then be integrated to obtain the temperature cross-section of the mixing layer. Raga & Cantó (1997) integrated this equation analytically with an idealized energy loss term. The resulting solutions demonstrate that narrow mixing layers are adiabatic (showing a parabolic temperature cross-section), and wide mixing layers are radiative (showing a flat-topped, quasi-isothermal temperature cross-section).
In order to obtain more concrete predictions from this model, we now compute a realistic radiative energy loss term. We consider the equations governing the fractional abundance f of each species. This abundance satisfies the equation:
where is the net sink term of the species (including collisional ionization, radiative and dielectronic recombination, charge transfer... and processes which populate the current species) and D the turbulent diffusivity which is assumed to be position-independent.
To complete the description of the mixing layer, we require lateral pressure equilibrium (which determines the density of the flow as a function of y), and calculate the turbulent viscosity with a simple, mixing length parametrization of the form:
where and are the density and sound speed (respectively) averaged over the cross-section of the mixing layer, h is the local width of the mixing layer (see Fig. 1), and is a constant. Considering that the turbulent conduction and diffusion Prandtl numbers are of order one, we can compute the conduction coefficient as (where is the heat capacity per unit mass averaged across the mixing layer cross-section) and the diffusion coefficient as .
The value was determined by Cantó & Raga (1991) by noting that this is the required value for a supersonic mixing layer model to match the opening angle of of subsonic, high Reynolds number laboratory mixing layers in the limit in which the jet Mach number tends to one. Also, these authors find that mixing layer models with this value for the constant reproduce the opening angle of mixing layers in jets with Mach numbers of up to 20.
value determined from fits to experimental results by Cantó & Raga 1991). Considering that the turbulent conduction and diffusion Prandtl numbers are of order one, we can compute the conduction coefficient as (where is the heat capacity per unit mass averaged across the mixing layer cross-section) and the diffusion coefficient as .
In this way, we obtain a closed set of second order differential Eqs. (2 and 4), which can be integrated with a simple, successive overrelaxation numerical scheme. The source terms for the atomic/ionic rate Eqs. (4) and the calculation of the radiative cooling (Eq. 2) were computed with the MAPPINGS code (described in detail by Binette et al. 1985; and Binette & Robinson 1987). We assume that the abundances of the elements included in the models (H, He, C, N, O, Ne, S, Fe) are solar.
In order to be able to compute solutions to the mixing layer problem, it is necessary to specify the width h of the mixing layer, the jet velocity , and the temperatures, ionization states and pressure of the jet and the surrounding environment.
The number of free parameters can be strongly reduced in the case of a wide, radiative mixing layer: In that case, the turbulent conduction term [first term of Eq. (2)] is only important close to the lateral edges of the layer. In the region which is not in direct contact with either the jet or the environment, the temperature of the mixing layer is determined by the balance between the viscous heating term and the radiative loss term. From this balance, and using the velocity gradient in Eq. (3), we obtain:
If the pressure of the gas is specified, this equation can be inverted to obtain the temperature of the mixing layer, which will be uniform (except for the lateral regions in which the turbulent conduction term is important) as a function of . Therefore, models with the same pressure and the same value of are approximately equivalent, except for regions close to the boundaries between the mixing layer and the jet/environment, which will not necessarily be identical for all models.
In cases where the layer is close to coronal equilibrium, the pressure can be combined with to yield a single free parameter. As radiative energy loss is dominated by low density regime line emission and recombination continua, we can write . Using the pressure balance condition, we then obtain from Eq. (6):
where is the environment pressure, for a monoatomic gas and the parameter has units of a surface density. The right hand side of Eq. (7) depends only on the temperature of the mixing layer. Hence, if coronal equilibrium holds, all models with the same value of are expected to be equivalent. This analytical result is qualitatively verified in our numerical calculations in the next section.
© European Southern Observatory (ESO) 1999
Online publication: May 6, 1999