Dr. Sergey Boronin School of Computing, Engineering and Mathematics Sir Harry Ricardo Laboratories

1. Optimal spatial disturbances of axisymmetric viscous jets Dr. Sergey Boronin School of Computing, Engineering and Mathematics Sir Harry Ricardo Laboratories University of Brighton

2. Outline • Introduction • Brief review of modal stability analysis • Key ideas of algebraic instability and optimal disturbances • Axisymmetric viscous jet in the air (main flow) • Formulation of linear stability problem for spatially-growing disturbances • Algorithm of finding optimal disturbances • Evaluation of jet break-up length based on optimal disturbances • Current issues/concerns

3. Linear stability analysis y Base plane-parallel shear flow: Small disturbances: Coefficients in governing equations depend on y only => Consider solutions in the form of travelling waves (equivalent to Fourier-Laplace transform) 1 U(y) x z 0 -1 q – vector of independent variables (normal velocity and normal vorticity for 3D disturbances), L – linear ordinary differential operator

4. Eigenfunctions and modal stability Temporal stability analysis:  - complex, kx, kz - real Eigenvalue problem: System of eigenfunctions (normal modes): {qn(y), n(kx, kz)} (discrete part of the spectrum, wall-bounded flows) + {q(y), (kx, kz)} (continuous part of the spectrum, open flows) Modal approach to the stability: Flow is stable  for a given set of governing parameters, all modes decay (Im{(k)} < 0, kx, kz)

5. Modal stability: pros and cons • Squire theorem (2D disturbances are the most unstable) • Modal theory predicts values of critical Reynolds numbers for several shear flows (plane channel, boundary layer) • Examples of failures: Poiseuille pipe flow (stable at any Re according to modal theory, unstable in experiments!) • Transition of shear flows is usually accompanied by 3D streamwise-alongated disturbances (”streaks”, see Fig.) 1 Alfredsson P.H., Bakchinov A.A., Kozlov V.V., Matsubara M. Laminar-Turbulent transition at a high level of a free stream turbulence. In: Nonlinear instability and transition in three-dimenasional boundary layers Eds. P.H. Duck, P. Hall. Dordrecht, Kluwer, 1996, P. 423-436. Fig. Visualization of streaks in boundary-layer flow1

6. Algebraic instability: mathematical aspect • A necessity for linear “bypass transition” theories (non-modal growth) • Mathematical reason for non-modal instability: • Linear differential operators involved are non-Hermitian (eigenvectors are not orthogonal) • Solution of initial-value problem is a linear combination of normal modes, non-exponential growth is possible (see Fig.) Fig. Time-evolution of the difference of two decaying non-orthogonal vectors (P.J. Schmid. Nonmodal Stability Theory // Annu. Rev. Fluid Mech. 2007. V. 39. P. 129-162)

7. Algebraic instability: lift-up mechanism y U(y) x Inviscid shear flowU=U(y) Consider disturbances independent ofx2: (linear growth, lift-up mechanism3) Inviscid nature, but still holds for viscous flows at finite time intervals! 2 M. T. Landahl. A note on the algebraic instability of inviscid parallel share flows // J. Fluid Mech. 1980. V. 98. P. 243-251 3 T. Ellingsen, E. Palm. Stability of linear flows // Phys. Fluids. 1975. V. 18. P. 487.

8. Optimal disturbances Expanding the disturbance of wave numbers kx, kz into eigenfunction series: (the set of coefficients {n} is aspectral projection of a disturbance q) Evaluation of the growth: density of the kinetic energy Disturbances with maximum energy at a given time instant t: (optimal disturbances)

9. Axisymmetric viscous jet in the air • Axisymmetric stationary flow • Both fluids (surrounding gas and jet liquid) are incompressible and viscous (Newtonian) • Cylindrical coordinate system (z, r, ) • Parameters of fluids: • (surrounding “gas” and jet liquid) • , are densities and viscosities • v, parevelocities and pressures • ( = g, l) gas  liquid r z

10. Non-dimensional governing equations Axisymmetric stationary flow: v = {u , v , 0}, /  = 0 (Reynolds numbers)

11. Boundary conditions At the infinity (r→ ∞): gas n liquid  Interface : H = r – h(z, t) = 0, n –normal unit vector: Kinematic condition at the surface: Continuity of velocity (no-slip):

12. Force balance at the interface Difference in stress at the surface is due to capillary force R: - Weber number and density ratio Kinematic condition at the axis r = 0 (all parameters should be finite)4: 4G.K. Batchelor, A.E. Gill, Analysis of the stability of axisymmetric jets. J. Fluid. Mech., 1962, V.14, pp. 529-551

13. Axisymmetric jet flow, local velocity profile • Assume that jet velocity profile varies slightly with z • (on the scale of wave lengths  considered) • For fixed z, consider “model ” axisymmetric solution: • cylindrical jet of radius r0(z): gas r0 liquid z z >> 

14. Linear stability problem Linearized Navier-Stokes equations for each fluid ( = l, g):

15. Normal modes Normal modes: Governing linear equations are reduced to analogues of Orr-Sommerfeldand Squire equations in cylindrical coordinates5: zero b.c. and conditions at the interface + 5D.M. Burridge and P.G. Drazin, Comments on ‘Stability of pipe Poiseuille flow’, Phys.Fluids, 1969,V.12, pp. 264–265

16. Solving eigenvalue problem Condition for nontrivial solution is a dispersion relation: Temporal and spatial analysis: • The goal is to find N normal modes with largest growth increments • Methods for solving dispersion relation directly are not efficient(e.g. orthonormalization method, result is a single mode, first guess is required!) • Reduction of differential eigenvalue problem to algebraic one is the most reliable • Eigenvalue kenters the governing equations non-linearly, reformulation of governing equations is needed • (addition of new variables, but reduction the order of k)

17. Reduction of the differential eigenvalue problem to algebraic one New variables: Governing linear equations: Normal modes: Eigenvalue problem: (L – 2nd-order linear differential operator in r)

18. Boundary conditions at r = 0, r → ∞ Gas disturbances decay at r → ∞: Kinematic condition at the axis r = 0 (all parameters should be finite):

19. Linearized boundary conditions at the interface Disturbed interface: • Boundary conditions are specified at perturbed interface (r = r0+h) and linearized to undisturbed interface r = r0: • Continuity • Kinematic condition • Force balance

20. Numerical solution of the eigenvalue problem • Finite-difference method (non-uniform mesh!) • reducing differential eigenvalue problem to algebraic eigenvalue problem for matrix – discrete analogue of differential operator L • QR-algorithm for the solution of algebraic eigenvalue problem (factorization into unitary and upper-diagonal matrices) • System of N normal modes (N is large enough)

21. Energy norm and optimal spatial disturbances • Energy norm: • Maximization of energy functional: (Ezis positive Hermitian quadratic form) Euler-Lagrange equations: (generalized eigenvalue problem for energy matrix) Optimal disturbances correspond to eigenvector with highest eigenvalue 

22. Possible application for break-up length evaluation Optimal disturbance growth is maximal in the spatial interval [0, z] • Threshold energy for break-up should be specified (experiments?) • Break-up of the jet with arbitrary disturbances occurs further upstream • Optimal break-up lengths provide lower-bound estimate for real jet break-up lengths at a given , m • Superposition of waves with different , m? 6M.I. Gavarini, A. Bottaro, F.T.M. Nieuwstadt, Optimal and robust control of streaks in pipe flow, J. Fluid. Mech, 2005, V. 537. pp.187-219 Example of optimal spatial growth (pipe flow)6

23. Current issues/concerns • Problem is formulated in the most general way. Possible simplifications? • Choosing the appropriate “local” jet velocity profiles Ug(r), Ul (r)? • Range of governing parameters of interest? • Evaluation of the jet break-up based on optimal perturbations?

24. Thank you for attention!