Numerical and asymptotic flow stability analysis of vortex structures

. Stability problem of an axisymmetric swirling flow of a viscous incompressible fluid with respect to nonaxisymmetric perturbations is considered. The system of ordinary differential equations for the amplitude functions is solved numerically by the Runge-Kutta method and orthogonalization procedure. Solutions of equations for perturbations at the neighborhood of singular points are obtained by the Frobenius method. The maximum of amplification coefficients and phase velocities of five unstable modes are calculated.


Introduction
Vortex flows are often observed in hydraulic engineering problems.The properties of the swirling flow are used in the suction tubes of hydraulic turbines, vortex spillways, countervortex energy absorbers, counter-vortex aerators, heat exchangers, purification structures, temperature and fractional separation devices [1][2][3][4][5][6][7].Flow stability analysis for the considered constructions is very important and actual problem.An effective tool for studying the hydrodynamic stability of a viscous incompressible swirling flows is a model based on the Navier-Stokes equations.The hydrodynamic stability problem of swirling flows for various configurations was studied numerically in [8][9][10][11][12][13][14][15].The results of experiments on the stability of swirling flows are presented in [16][17][18].In this paper, an effective method for calculating the stability of a swirling flow with an arbitrary initial velocity profile is presented.

Problem formulation
Let's consider an axisymmetric viscous incompressible flow in a cylindrical coordinate system ) , , ( z r M .The system of the Navier-Stokes equations for the variables velocity and pressure is written in the form: where are the axial, radial and azimuthal velocity components, p is the pressure, Re is the Reynolds number.Suppose that the flow under consideration has the following velocity distribution: We will investigate the behavior of small perturbations for flow (5) in the form: where D is the wave number; n is the disturbance mode ;...) 2 ; 1 ; 0 ( r r n , c is the wave propagation velocity; i is the imaginary unit; are the complex amplitude functions. After linearization equations ( 1)-( 4) and substitution expressions ( 5), (6), we obtain the ordinary differential equations system for the complex amplitude functions: where . The boundary conditions at 0 r for system ( 7)-( 10) can be written in the form: The conditions on the pipe wall at 1 r have the form: In this study, perturbations (6) are considered periodic in z with a time-varying amplitude.So D is a real value ( , where O is the perturbation wavelength), and is a complex value, r c is the propagation disturbance velocity in z direction (phase speed), i c is the growth rate disturbance in time.The amplitudes of the disturbance (6) decay (the flow is stable), if 0 i c , and grow with time (the flow is unstable), if 0 !i c .

Numerical method
We represent system ( 7)- (10) in matrix form: The resulting system (16) has a regular singular point at 0 r .In the general case, a linear system of differential equations has a regular singular point if we can write the original system in the form: and all functions at the point 0 r are regular and do not vanish simultaneously.The solution can be found by the Frobenius method in the power series form: where O is the root of the characteristic equation q p q p e e f q p q p q p Let's use the Frobenius method to find the solution in this case.We write system (16) in the form: where matrices k A consist of coefficients at powers k r .We will find the solution in the form: Substituting it into the system of equations ( 16), we obtain: Equating the coefficients at the same degrees, we obtain the following matrix system for vectors: where E is the identity matrix.The characteristic equation allows you to find O : . Three of them should be discarded, since the perturbations should be bounded at 0 r .We will number the remaining three values in such a way that the condition is satisfied.After finding the expansion coefficients, we get the fundamental solutions system.To determine the coefficients, a computer program was developed that allows to calculate any number of terms in the expansion (18).Finding the coefficients is reduced to solving systems of linear equations.
The stability problem for swirling flows unbounded by solid walls has one more singular point at f o r . To find a solution in this case, we write the system of equations ( 7)- (10)   .The solution at this irregular singular point will be sought in the form To determine the expansion coefficients, it is necessary to solve the system of matrix equations: etc , ) ) Here Q B is the matrix consisting of the expansion coefficients of the matrix

Calculation results and discussion
The considered asymptotic method allows us to find a solution for the system of equations ( 7)- (10) in the neighborhood of singular points.Using these expansions, the integration of equations ( 7)-( 10) was performed by the Runge-Kutta method with an automatic choice of the step.To ensure the stability of calculations, the Gram-Schmidt method was applied.The software package implementing this algorithm was compiled in the Fortran-90 programming language.
As a test problem, we investigate flow stability in a tube rotating around its axis at constant angular velocity q : ( The calculated dispersion curves for a fixed Reynolds number 1000 Re and 1 q are shown in Figure 1.In this case, there are five unstable modes.All these modes are inviscid, since the amplification coefficients tend to some constant values at  The developed algorithm makes it possible to investigate the stability of swirling flows with an arbitrary velocity profile and to determine the range of the fastest growing disturbances.Analysis of the calculated dependences contributes to the correct choice of the optimal operating mode of vortex devices in order to avoid the phenomenon of vortex destruction. The work was supported financially by the Russian Foundation for Basic Research (project No 18-01-00762).

Fig. 1 .
Fig. 1.Amplification coefficients (solid lines) and oscillation frequency (dashed lines) at the vector of the required coefficients.From the first