1. Numerical study of flow instability between two cylinders in 2D case V. V. Denisenko Institute for Aided Design RAS

2. Investigation of flows stability between coaxial cylinders has besides fundamental interest, also great practical sense, because such flows often occur in different technical equipments. Problem definition. The mathematical model is based on model of inviscid compressible gase and it includes integral laws of conservation of mass, energy and momentum. The system is closed by equation of state of ideal gase. Supposed, that Reunolds number (numerical Reunolds) sufficiently great, so that flow can be unstable. In the case of initial data we take inviscid Couette flow. In the middle of clearance between cylinders the local perturbation of radial component of velocity with small amplitude and defined value of frequency is brought. On the boundary we used nonflow conditions. This statement is based on two hypotheses, that finding their confirmation in experiments: 1.Independence of large-scale ordered structures of turbulent flow and small-scale stochastic turbulence for great Reunolds numbers; 2.Weak influence of viscosity on developing large-scale structures;

3. Numerical experiment. The numerical modeling is made with TVD method. We took the polar grid with dimension, where - number of nodes by radius, - number of nodes by angular. In numerical experiment the turbulent energy was calculated, where- the kinetic energy of flow at initial time, - pulsations of radial and angular component of velocity respectively. Pulsations was calculated like that: averaging both components on angular was being making, with formula where- average function value. Then pulsations were being calculating by formula: The investigation was being performing on several parameters of task: difference of velocities of internal and external cylinders, width of clearance between cylinders, amplitude and frequency of perturbation. In the case of investigation of task on difference of velocities, internal cylinder was resting, external was rotating. All calculations were being making on grid with dimension 51x350. In scheme attends the dissipative mechanism, associating with scheme viscosity.

4. Influence of grid dimension We consider, how grid influences on experiment results. For this four experiments had been made for cases of difference of angular velocities and. Calculations were being making on grid with dimensions Here the graphs of turbulent energy in dependence from time is showed.. Graph for case and grid 77-525 Energy of developed turbulence Time of turbulence is begin 77x525 и 101x700.

5. and grid 101x700 and grid 77x525

6. grid 101x700 From graphs we can see, that in the case of and grid 77-525=35.6, =0.0146; atand grid 101-700=34, =0.0195. To velocity and grid 77-525 =23, =0.0382, in the case of grid 101-700 =26,=0.0416. It is obvious, that the energy of turbulent flow increases with growth of grid dimension, this is linked with that the value of grid viscous increases and the dissipation of energy occurs with smaller velocity.

7. This figure illustrates independence of large structures scales from grid dimension Grid 51-350 Grid101-700 Grid 77-525 ( case )

8. Experiment results. Influence of difference of velocities between cylinders Difference of angular velocities Difference of angular velocities Let go to investigation of difference velocity influence of internal and external cylinders. In problem statement the internal cylinder is still, so that only external cylinder is rotating. Clearance width we take equal to, and radius At this figure vorticity distribution at calculation start is showed. Perturbation (11 modes of wave) streamlines of middle of clearance

9. On this figure we can see vorticity ring forming (inflection area of vorticity or it maximum). We will see below, that from this area turbulization of all flow is begun. Vorticity ring

10. At end of calculating time from inflection vorticity area vortexes is formed, which scale smaller than width of clearance between cylinders, so we can say, that flow proceeds in weak turbulent state. Small-scale vortexes

11. This figure illustrates dependence of turbulent energyfrom time t. turbulization start, time t’~15 Average energy of fully developed turbulence

12. Difference of angular velocities Next we twice multiply the angular velocity of external cylinder. In this case also inflection vorticity area is formed (or it maximum). Inflection area of vorticity

13. Here we see figure of vorticity to flow turbulization start. We can see, that vortexes is formed from vorticity maximum area. Forming vortexes.

14. Formed, vortexes begin to actively interact among themselves (to pair) and to end of time calculation three vortexes have been remained, with scale about width between cylinders. Vortexes pairing.

15. From graph of turbulent energy we see, that the initial time of flow turbulization t’~25, energy of turbulent flow. Turbulization is begin. Energy of turbulent flow. Thus, with previous case (half as great velocity), the transition in pulsating regime have much more expressed character: the energy of turbulent flow with the scale of vortexes has been increased.

16. Difference of angular velocities Now we twice increase the velocity of external cylinder. As in previous two cases, in this case vorticity ring is formed and from this ring vortexes is arisen.

17. At end of the calculation time, as previous case, three vortexes with scales about clearance between cylinders are remained.

18. The initial time of flow turbulization t’~18 is decreased in comparison with previous case, energy of turbulent flow is increased to

19. Thus, we obtain, that so bigger difference of velocities between cylinders, that smaller the time of turbulization and greater the energy of turbulent flow. This is explained that so greater differ of velocities, than greater pressure gradient between internal and external cylinders. Pressure gradient generates the moment of forces, under which vortexes is born, and so greater this gradient, than early vortexes is born. More specifically, the type of instability, occuring at current statement of problem (Couette flow is being investigated), has «shear character». Shear lay here extends to all clearance between cylinders. Initial profile line of velocity

20. Influence of clearancewidth Influence of clearance width Now we change width of clearance, leaving the average radius of channel fixed. The calculations is carried out for interval =[1.0;0.1] with step 0.1, the perturbation we leaved without change. The flow character no change in comparison with previous cases: in the beginning the distribution of vorticity near uniformly, then, in that range, where the perturbation had been injected, the vorticity ring is formed and from this ring the vortexes are born and the flow turbulization is begun. We show here only the graph of dependence of time begin of turbulization t’ from width of clearance, because all the figures of bearing and evolution of vortexes similar to considered above and not have special interest.

21. Here at=0.1 the time t’ is put equal to 50, but the computation is performed up to time ~ 30 and the flow is not become turbulent. From this graph we see, that we have minimum, occuring to The occurrence of this minimum is explained, that for right side of minimum the width of clearance increase and pressure gradient (per unit length in radial direction) in area, where the vorticity ring is formed, decrease. Thus, the moment of forces, under which the vortexes are born, is decreased. Therefore, the time of flow turbulization is increased.

22. Increasing the time of flow turbulization on the left of minimum is explained by influence the clearance wall, which don’t give the vortexes with scale greater than width of clearance to bear. From experiments we may observe, that the number of initially bearing vortexes have value about the number of mode of wave perturbation. From this it follow, that the more the mode of wave, than the greater number of vortexes the ring will be broken down and than less will be scale of burning structures. Therefore, increasing the number of mode of wave perturbation would to move the minimum at coordinate origin. Here the dependency of energy of fully developed turbulence from the clearance width is showed.

23. Influence of perturbance amplitude In experiments the values of perturbances amplitudes is supposed equal to 0.01, 0.02, 0.05, 0.08, 0.1, 0.15, 0.2. We show here the graphs of energy of turbulent flow and time of turbulization begin t’ from amplitude а. We see, that the turbulent energy is weakly depended on amplitude

24. Time of flow turbulization also is weakly depended on а. Thus, we can say, that perturbance amplitude does not influence on flow character (weakly influences as compared with other parameters).

25. Influence of perturbance frequency Now we go to study of character influence of perturbance frequency. We take clearance width =0.5 and amplitude of perturbance about 4% to flow velocity. Also, how it was in two previous cases we show here graphs of turbulent energy of fully developed turbulence and initial time of flow turbulization t’ from frequency n. The values of mode wave numbers of perturbations we changed in interval from 3 modes of wave to 22 by step to one number mode of wave. The energy of turbulent flow is increased by mode of wave number n is grown.

26. From this graph we see, that t’ is increased by mode of wave number is grown. Thus, we obtain the long wave instability, i.e. the long waves more unstable, that short waves. Because the mode number of waves is about number of born vortexes, then that greater the wavelength, than bigger vortexes are born and, respectively, the moment of force, under which they are born, become greater. This may be cause of long wave instability.

27. Conclusion In this work was carryed out the numerical experiment to investigation Couette flow instability between two cylinders. The flow instability is revealed, which is explained it «shear character».Influence of different task parameters is investigated, so that: difference of velocities external and internal cylinders, width of clearance between cylinders, amplitude and frequency injected perturbance. With increasing difference of velocities between cylinders, the initial time of flow turbulization t’ is decreased, that is linked with growth pressure gradient to radial direction and, respectively, growth moment of force, which brings to vortex bearing. Vortexes are born from inflection vorticity area, which is arisen there, where perturbations was injected. The dependencies graphs of time of initial flow turbulization and fully developed turbulence energy was showed at cases influence parameters, a и n. From which we can see, that amplitude weakly influences on flow character, the clearance width influences via cylinder walls and changing of pressure gradient per unit of distance. In the case of influence frequency the longwave instability is occured.