Stability of counter vortex flows in hydraulic engineering construction

In the framework of linear theory, the stability of counter vortex flows with respect to non-axisymmetric perturbations is investigated numerically. The main flow field calculation results have been obtained as the solutions of the Navier-Stokes equations. The amplification coefficients are calculated, the regions of instability of the flow are defined.


Introduction
Counter vortex flows are formed in the interaction of two (or more) coaxially rotating flows, swirled in mutually opposite directions. In technical applications, counter vortex flows are used in combustion chambers, gas turbine engines, heat exchangers, bioreactors, fermenters, cooling towers, and other technical devices [1][2][3].
In hydrotechnical construction, counter vortex flows are used in vortex spillways to quench the energy of high-speed water flows [4], in counter-vortex aerators [5,6] to create two-phase water-air flows. A detailed classification of devices using the interaction effect of oppositely swirling flows of liquid and gas is presented in [7].
The hydraulic characteristics of counter vortex flows were studied in [8]. An analytical study of the main flow field based on the Oseen model, is given in [9]. The study of local stability of swirling flows using the Rayleigh number criterion is presented in [10].
This paper is devoted to numerical modeling of hydrodynamics and stability of counter vortex flows using the full system of Navier-Stokes equations to calculate the main flow. The stability of the swirling axisymmetric flows is considered on the assumption of local parallelism using numerical method: the problem of the normal modes developing against the background of the axisymmetric flow determined by the velocity profiles in local cross sections of the flow is solved.

Hydrodynamic flow model
We will consider a viscous incompressible flow in an axisymmetric channel with solid impermeable walls. Internal swirling flow is fed to the central part of the channel ( ). The present analysis is based upon the numerical solution of the full Navier-Stokes equations. In the cylindrical coordinate system z r , , the Navier-Stokes equation can be represented in terms of the stream function  , the vorticity  and azimuthal velocity z V in form: System (1) -(4) is presented in conservative form and contains the two dimensionless parameters: the swirl number The function ) (r f is defined from the initial distribution of the axial velocity for 0  r according to (4). The initial velocity distribution is defined as: The values of the coefficients in expressions (6) To solve numerically the boundary value problem (1)-(4), (6)-(10) the finite difference method [11] was used with a uniformly spaced grid 81x257 for 30  500 . The total number of cases calculated was about 200. The most important properties of the flows are associated with the development of recirculation zones in the near-axis and near-wall sections in the neighbourhood of the tangential swirler. The most characteristic streamline patterns const   are shown in Fig. 1. Comparison of the numerical results with the experimental data [12] for turbulent flows in a vortex chamber with oppositely rotating flows is given in [11]. Good agreement between the results was obtained using a turbulent analogue of the Reynolds number, calculated from the turbulent viscosity t  using the following expression where  is the hydraulic drag coefficient,  is the universal constant. The coefficient  varies in the range of 0.011-0.03. The values of the universal constant  are 0.2, 0.07 for water and air, respectively. In this case, the turbulent Reynolds number is 80-135 for water and 230-385 for air. Therefore, the calculations performed in the selected range of Reynolds numbers allow us to correctly describe the flow hydrodynamics. For all cases considered in [12], the axial velocity distribution on the flow axis, the diameter of the reverse zone and the counter flow rate are in good agreement with experimental data. 8 (a-c); 250 Re  ; (d-f). Let us consider small perturbations of the traveling wave type (normal modes) for the calculated flows, determined by the profiles of the axial ) (r U and the azimuthal ) (r W velocity components in local flow cross sections Here p is the pressure,  is the wave number, n is the perturbation mode ;...) 2 ; , (the positive values of n correspond to the wave propagation in the direction of swirl, whereas the negative ones correspond to that in the opposite direction), c is the wave speed, i is the imaginary unit. Then, for the complex-valued amplitude functions we obtain the following system of equations: Here is the Reynolds number, 0 U is the axial velocity for 0  r , L is the vortex core radius corresponding to the maximal value of ) (r W , and the prime indicates the derivative with respect to r . Assuming that the solution near the axis 0  r is regular, we come to the following boundary conditions for (13)-(16): , We consider the perturbations (12) for which 0   is a real number. In this case, an eigenvalue with Newton's method used for the corresponding characteristic equation. We used this method in [13][14][15] to analyze the stability of swirling flows of various types.
The calculation examples of problem (13)-(16) with conditions (17)-(20) are presented at Fig. 2. The flow stability was investigated for perturbations (12) with 1   n , since, according to [16][17], this mode is the most dangerous for both swirling pipe flow and for free vortices, while in [18] it is shown that it is probably precisely this mode that is observed in the experiments.

Conclusions
The mathematical model used to study the hydrodynamics of the counter-vortex flows allows one to fairly accurately describe the structure of swirling flow with the formation of axial recirculation zones.
Considering the local cross sections, we find that as z increases the flow instability first rises and then falls. Thus it is possible to identify a certain region of instability bounded with respect to z and possessing the following properties: for fixed swirl an increase in the Reynolds number amplifies the flow instability, and the region of instability itself grows larger; for fixed Re an increase in swirl leads to only a slight upstream displacement of the boundary of the region of instability; when a reverse flow zone is present, the strongest instability is observed in that zone.
Increasing the swirl of the external flow has a stabilizing effect on the counter vortex flow in all the considered cases.
The work was carried out with financial support from the Russian Foundation for Fundamental Research (project No 18-01-00762).