1. Absolute and convective instability of the two-dimensional wake
EFMC6 Stockholm, June 26-31, 2006
Absolute and convective instability of the two-dimensional wake
D.Tordella #, S.Scarsoglio # and M.Belan *
# Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di
Torino, Italy
* Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di
Milano, Italy

2. Near-parallel flow assumption
STATE OF THE ART
Near-parallel flow assumption
Important feature of most of the linear stability analyses in literature (Mattingly and Criminale, JFM 1972; Triantafyllou et al., PRL 1986; Hultgren and Aggarwal, PoF 1987);
Sequence of equivalent parallel problems: at each section behind the cylinder the properties of the wake are represented by the stability properties of a parallel flow having the same average velocity profile;
Lateral convection of the base flow neglected;
Instability originates in the near wake, where the streamlines are not parallel. The disturbances grow linearly in a region of absolute instability, that is downstream to the back stagnation point of the body and which is preceded and followed by convectively unstable regions (e.g. Monkewitz, PoF 1988; Young & Zebib, PoF 1989; Pier, JFM 2002)  global linear instability as the critical value of the flow control parameter is exceeded;
The nature of the instability is related to the dominant saddle points of the local complex dispersion relation.

3. In this study, the Reynolds number is the only parameter.
◆ Basic flow previously derived from intermediate asymptotics [Tordella &
Belan, PoF 2003, ZAMM 2002].
◆ Multiscaling is carried out to explicitly account for effects associated to the
lateral momentum dynamics at a given Reynolds number.
◆ These effects are an important feature of the base flow and are included in the
perturbative equation as well as in the associated modulation equation.
■ At the first order in the multiscaling, the disturbance is locally tuned to the
property of the instability, as can be seen in the zero order theory (near-parallel
parametric Orr-Sommerfeld treatment).
■ Synthetic analysis of the nonparallel correction of the instability characteristics. The system is locally perturbed by waves with a wave number that varies along the intermediate wake and which is equal to the wave number of the dominant saddle point of the zero order dispersion relation, taken at different Reynolds numbers [Belan & Tordella JFM 2006; Tordella, Scarsoglio, Belan PoF 2006].
In this study, the Reynolds number is the only parameter.

4. Basic equations and physical problem
Steady, incompressible and viscous base flow described by continuity and Navier-Stokes equations with dimensionless quantities U(x,y), V(x,y), P(x,y), c
R =cUcD/
Boundary conditions:
symmetry to x, uniformity at infinity and field information in the intermediate wake

5. To analytically define base flow, its domain is divided into two regions both described by Navier-Stokes model
Inner region flow -> ,
Outer region flow -> ,
Pressure longitudinal gradient, vorticity and transverse velocity are the physical quantities involved in matching criteria.
Composite expansion is, by construction, continuous and differentiable over the whole domain.
Accurate representation of the velocity and pressure distributions - obtained without restrictive hypothesis - and analytical simplicity of expansions.

6. vc vo vi uo ui uc
po - p
pc - p
R = 34, x/D = 20. Fourth order of accuracy – Inner, outer and composite expansions for velocity and pressure.
pi - p

7. Linear stability theory
Base flow is excited with small oscillations.
Perturbed system is described by Navier-Stokes model
Subtracting base flow equations from those concerning perturbed flow and neglecting non linear oscillating terms, the linearized perturbative equation in term of stream function is
Normal modes theory
Perturbation is considered as sum of normal modes, which can be treated separately since the system is linear.
complex eigenfunction,
u*(x,y,t) = U(x,y) + u(x,y,t) v*(x,y,t) = V(x,y) + v(x,y,t) p*(x,y,t) = P0 + p(x,y,t)

8. h0 = k0 + i s0 complex wave number s0: spatial growth rate
k0: wave number
h0 = k0 + i s complex wave number s0: spatial growth rate
0 = 0 + i r complex frequency 0: frequency
r0: temporal growth rate
Perturbation amplitude is proportional to
r for at least one mode unstable flow
r for all modes stable flow
s for at least one mode convectively unstable flow
s for all modes convectively stable flow
Convective instability: r0 0 for all modes, s0 0 for at least one mode. Perturbation spatially amplified in a system moving with phase velocity of the wave but exponentially damped in time at fixed point.
Absolute instability: r (vg=0/h0=0 local energy increase) for at least one mode. Temporal amplification of the oscillation at fixed point.

9. Stability analysis through multiscale approach
Slow spatial and temporal evolution of the system slow variables x1 = x, t1 = x.
= 1/R is a dimensionless parameter that characterizes non-parallelism of base flow.
Hypothesis: and φ(x, y, t) are expansions in term of :
By substituting into the linearized perturbative equation, one has
(ODE dependent on ) + (ODE dependent on , ) + O (2)
Order zero theory. Homogeneous Orr-Sommerfeld equation (parametric in x1 ).
where , and A(x1,t1) is the slow spatio-temporal modulation, determined at next order.
By numerical solution eigenfunctions 0 and a discrete set of eigenvalues 0n

10. First order theory. Non homogeneous Orr-Sommerfeld equation (x1 parameter).
is related to the zero-order dispersion relation, the base flow, and explicitly considers non-parallel effects through the presence of the transverse velocity

11. To obtain first order solution, the non homogeneous term is requested to be orthogonal to every solution of the homogeneous adjoint problem, so that
Keeping in mind that , the complete problem gives
First order corrections h1 e 1 are obtained by resolving numerically the evolution equation for modulation and differentiating numerically a(x,t) with respect to slow variables.

13. 0(k0,s0) - R = 35, x/D = 4.

14. r0(k0,s0) - R = 35, x/D = 4.

16. Frequency. Comparison between the present solution (accuracy Δω = 0
Frequency. Comparison between the present solution (accuracy Δω = 0.05), Zebib's numerical study (1987), Pier’s direct numerical simulations (2002), Williamson's experimental results (1988) .

17. Eigenfunctions and eigenvalues asymptotic theory
An asymptotic analysis for the Orr-Sommerfeld zero order problem is proposed. For x   the eigenvalue problem becomes
where
This analysis offers a priori inferences in agreement with the far field results yielded by the numerical integration of the modulation equation.
Comparison between the asymptotic curve (with unitary proportionality constant) and the saddle point curve r0(x).

18. Disturbance tuned to the local physical wave numbers along the wake.
CONCLUSIONS
The multiscaling explicitly accounts for the effects associated to the lateral momentum dynamics, at a given Reynolds number.
Disturbance tuned to the local physical wave numbers along the wake.
The first-order correction allows absolute instability pockets to be determined in the first part of the intermediate wake (up to x ~ 12). No overlapping with the simmetric counter rotating vortices.
The far-wake asymptotic behavior independently obtained through an analysis based on the properties of the Orr-Sommerfeld operator.
These pockets are present when the Reynolds number R is equal to 50 and 100, but are absent when R is as low as 35. This is in general agreement with the standard notion of a critical Reynolds number of about 47 for the onset of the first observable instability.
All the four instability characteristics vanish at infinity downstream to the body creating the wake flow.

19. REFERENCES
Tordella, D; Scarsoglio, S; Belan, M
A synthetic perturbative hypothesis for multiscale analysis of convective wake
instability
PHYSICS OF FLUIDS, 18 (5): Art. No MAY 2006
Belan, M; Tordella, D
Convective instability in wake intermediate asymptotics
JOURNAL OF FLUID MECHANICS, 552 : APR
Tordella, D; Belan, M
On the domain of validity of the near-parallel combined stability analysis for
the 2D intermediate and far bluff body wake
ZAMM, 85 (1): JAN 2005
A new matched asymptotic expansion for the intermediate and far flow behind
 a finite body
PHYSICS OF FLUIDS, 15 (7): JUL 2003
Asymptotic expansions for two dimensional symmetrical Laminar wakes
ZAMM, 82 (4):

20. Inner and outer expansions – Details up to third order

21. Order 0

22. Order 1

23. Order 2

24. Order 3
C3 to be determined with boundary conditions in x=x*
where

25. Conclusions and further developments
Analytical description of the base flow
Perturbation hypotesis – Saddle point sequence
Spatio-temporal multiscale approach
Comparison with numerical and experimental results
Asymptotic analysis for the far wake
Entrainment
Validity limits for first order corrections

26. #