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Concept of hydrodynamic
stability
Contents:
1.
2.
3.
4.
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8.
9.
History of the problem
Instability
Modern research
Reynolds experiment
Examples of some other instabilities
Concept of hydrodynamic stability
Critical numbers for the onset of instability
Equation of energy balance
References
History of the problem
Perhaps the first ever vortex
‘visualization’ by Leonardo da Vinci
How flows become disturbed?
Instability
Simple mechanical examples of equilibrium states: a, stable state; b, unstable
state; c, neutral (indefinite) state.
For certain parameters of hydrodynamic system, the hydrodynamic equations of motion with
precise stationary (laminar) solutions cannot be implemented in practice: the motion is
unstable. Consequently, an important subject in the theory of hydrodynamic stability is the
analysis of the development of disturbances in an initially laminar flow.
Modern history of shear flow
stability and transition research
Some milestones:
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Reynolds pipe flow experiment (1883)
Rayleigh’s inflection point criterion (1887)
Orr (1907) Sommerfeld (1908) viscous eq.
Heisenberg (1924) viscous channel solution
Tollmien (1929) Schlichting (1933) viscous
BL solution
Schubauer & Skramstad (1948)
experimental TS-wave verification
Klebanoff, Tidstrom & Sargent (1962) 3D
breakdown
…
Reynolds experiment
At certain flow parameters a laminar flow cannot remain stationary –
disturbances lead to dramatic changes in it. The subject in the theory of
hydrodynamic stability is the analysis of the development of the
disturbances.
Reynolds experiment 1883:
Dye into center of pipe
The flow is laminar (small velocity)
Transitional flow (intermediate velocity)
Developed turbulence (large velocity)
Examples of some other instabilities
Benard convection at
a heated plane surface
Taylor vortices in flow between
two rotating cylinders (Couette
flow)
Illustration of flow instability (video)
Click to play
Video of Yu. A. Litvinenko,
G.R. Grek, V.V. Kozlov and G.V. Kozlov (2010)
Concept of
hydrodynamic stability
• To be used in hydrodynamic applications, the definition of stability must be
properly specified. Because of the complexity of the hydrodynamic
equations of motion, it is obviously not possible to give a unique rational
definition of the stability. In general, if at importation of certain
disturbances in a flow, it returns to its initial state (speaking in the language
of the theory of dynamic systems, is ‘attracted’) in time and/or space, the
flow is stable to these disturbances.
• Frequently in the stability problems the asymptotic (after long period)
response of a system affected by a disturbance is considered. However,
situations are not excluded where the disturbance at the beginning, during
its establishment, experiences transient growth and only then decays.
• Transient phenomena can be found in many other branches of physics, such
as at the breaking of an electric circuit that can lead to a bubble break.
Electrical analogy with
transient growth
bulb
peak voltage
Voltage
supply
vout
working voltage
capacitor
Transient voltage beatings
A simple RC circuit.
Let's consider a circuit having something other than resistors and sources. We know that
vin=vR+vout. The current through the capacitor is given by
dv
I  C out
dt
and this current equals that passing through the resistor. Using vR=IR gives
dvout
 vout  vin
The input-output relation for circuitsdtinvolving energy storage elements takes the form
RC
of an ordinary differential equation, which we must solve to determine what the output
voltage is for a given input.
Similarly, if a transient disturbance in a hydrodynamic flow becomes dangerously large
during this transient growth, it can trigger the laminar–turbulent transition.
Classes of physical problems regarding the
propagation of disturbances in hydrodynamic
systems
• The problem of initial conditions or stability in time. If the
initial disturbance decays in time at each fixed point of space
(or, at least, does not monotonically grow), the system is
called stable to these disturbances. Otherwise, if the initial
disturbance monotonically grows in time at a fixed point of
space, the system is called absolutely unstable. In physiscs
such systems are denoted sometimes as “generators”.
In the case of the electric current
there is only one independent
variable (time). In
hydrodynamics there are four:
x,y,z,t.
Disturbance source
’
Classes of physical problems regarding the
propagation of disturbances in hydrodynamic
systems
•The problem of boundary conditions or amplification
in space. If an external signal at the entrance to the
system decays whilst propagating in it, it is said that the
spatial attenuation (non-transmission of a signal) takes
place. Otherwise, there is a spatial amplification, and
the system is called convectively unstable. In physics
such systems are called sometimes “amplifiers”.
Disturbance source
Critical numbers for the onset of
instability
• The instability is defined as a state when the corresponding stability conditions
are broken. In accordance with the principle of similarity first discovered
experimentally by Reynolds (1883) for flow in a channel, the conditions for the
appearance of the instability depend on certain dimensionless ratios of problem
parameters as viscosity, density, velocity, temperature and frequency. For many
simple flows, such a fundamental quantity is the Reynolds number Re=Ul/ν.
Based on the definitions of stability given above, the critical Reynolds numbers
separating the regions of stable and unstable motion are defined below.
• For more complex flows, such as those with curvature, other similarity
parameters, e.g., the Görtler number Gö, which is a dimensionless measure of a
curvature of a wall, appear. Then the same classification is applicable to them
as well.
Stability of fluid motion in time
• The concept of stability in time can be defined using various positivedefinite norms of parameters (measures) of the disturbances.
• The natural physical measure of the disturbance is usually its kinetic
energy. Therefore, we give various formal definitions of the stability based
on the kinetic energy of disturbance velocity u, integrated over the whole
volume V covered by the hydrodynamic system
2
u
EV   dV
2
V
• This implies either a localization of the disturbance in the volume V which
is large enough for open flows (the disturbance developing only inside the
volume during the observation) or the spatial periodicity of the motion for
closed flows, V covering the whole range of the disturbance motion.
Definitions of stability
• Asymptotic stability: A flow is (asymptotically) stable to disturbances, if
limt 
EV ( t )
 0,
EV (0)
where t is time.
• Conditional stability: If there is a value δ > 0, such that any solution of
equations of motion is stable at E(0) < δ, the solution is called conditionally
stable. The value δ (the attraction radius) determines a set of initial conditions
attracting to the undisturbed solution. If the disturbance energy E(0)≥δ, the
disturbance grows or forms a new stable state (exchange of stability).
• Global stability: If the value δ→∞, the solution is globally or unconditionally
stable.
• Monotonic stability: If the flow is globally stable and dE/dt ≤0 at all t > 0, the
solution is monotonically stable.
As seen, each next definition imposes new restrictions on the stability.
• At Re>ReE the flow loses monotonic stability; i.e. the
disturbances are possible, whose energy can experience a transient
growth.
• At Re>ReG the flow loses global stability – it becomes
conditionally stable. The sense of the conditional stability is easily
seen from the simple mechanical model: the system is stable to
infinitesimal disturbances, but unstable to disturbances exceeding
a certain threshold value in amplitude (the ‘condition’). This is in
contrast to the ‘global’ stability case.
• In other words, at Re>ReG there may be initial disturbances
capable, as a minimum, not to decay in time and, as a maximum,
cause transition to turbulence. For such flows, where the principle
of exchange of stability holds, the transition Reynolds number
ReT obviously differs from ReG, i.e. stable-not-turbulent
equilibrium solutions have to be taken into account.
Schematic view of the different Reynolds number regimes
based on the preceding definitions
I. Monotonic decay (all disturbances decay)
II. Global stability (some disturbances may grow, but will
decay, as time evolves.
III. Conditional stability region (only disturbances with
energy more than certain value can grow without further
decay)
IV. Linear instability region (stable flow cannot exist at all)
At Re>ReL the flow is linearly unstable, i.e. there is an infinitesimal
disturbance which does not decrease in time. The plane Poiseuille flow and the
Blasius boundary layer becomes linearly unstable at certain finite Reynolds
numbers. However, there are flows such as the flow in a round pipe, which are
linearly stable at any Re. [‘Linear’ means that due to the smallness of the
disturbances all non-linear terms in the NS equations can be dropped.]
As we will see below, global stability relates closely to the linear stability of
small disturbances for the flows of our interest.
Critical Reynolds numbers for a
number of wall-bounded shear flows
Flow
Pipe
ReE
ReG
ReT
ReL
81.5
-
2000
∞
49.6
-
1000 5772
20.7
125
(Hagen-Poiseuille flow)
Channel
(Poiseuille flow)
Moving
walls
(Plane Couette flow)
Re 
Ul

360
∞
Equation of energy balance
(Reynolds-Orr equation)
•The equation of energy balance is one of the consequences of the NavierStokes equations for disturbances. It is obtained by multiplication of the
nonlinear disturbance equation
u
1 2
 (U)u  (u)U  (u)u  p 
u
t
Re
by u and integration by parts over volume V using continuity equation
(u )  0
The procedure yields:
Equation of energy balance
(Reynolds-Orr equation)
In compact form it can be written as
where
dEV
1
2
   (uD )udV 
(

u
)
dV

dt
Re V
V
 U
U j 
D i 
/2
xi 
 x j
is the symmetric deformation tensor of the mean laminar flow.
As seen, the non-linear terms vanish in the equation. Due to multiplication
by ui linear terms in Navier-Stokes equations correspond to quadratic
terms in the equation of the energy balance.
This indicates that the possible energy growth of any disturbances in
incompressible Newtonian fluids is related to linear growth mechanisms.
Equation of energy balance
(Reynolds-Orr equation)
The first right-hand side term in the equation
dEV
1
   (uD)udV 
(u) 2 dV

dt
Re V
V
always negative
describes the exchange of energy with the mean flow, the second term
describes the energy dissipation due to viscosity. If the exchange term is
positive, the energy is extracted from the mean flow. The dissipation term is
always negative. The relative value of these two terms defines whether the
energy of the disturbance decays or grows. The rate of these two terms at
dEV/dt = 0 forms the global stability problem for ReE, below which any
disturbance monotonically decays.
This can be formulated as a variational problem to search for allowable
velocities which maximize 1/ReE – the rate of two quadratic forms (the
Rayleigh quotient):


(uD)udV
 V

1
 max

Re E u ( r )0   (u )2 dV 
 V

It is known that such a maximum always exists and is positive.
References
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Reynolds O. (1883) An experimental investigation of the circumstances which determine whether
the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels,
Philos. Trans. R. Soc. Lond. A, Vol. 174, pp. 935‒982.
Lord Rayleigh (1987) On the stability, or instability, of certain fluid motions, Proc. Lond. Math.
Soc., Vol. 19, pp. 67‒75.
Orr W. M'F. (1907) The stability or instability of the steady motions of a perfect liquid and of a
viscous liquid. Part 2: A viscous liquid, Proc. R. Irish Acad. A, Vol. 27, pp. 69‒138.
Sommerfeld A. (1908) Ein Beitrag zur hydrodynamischen Erklärung der turbulenten
Flüssigkeitsbewegungen, Atti IV Congr. Internaz. Mat., Vol. 3, pp. 116‒124.
Heisenberg W. (1924) Über Stabilität und Turbulenz von Flüssigkeitsströmen, Ann. Phys., Vol. 74,
pp. 577‒627.
Tollmien W. (1929) Über die Entstehung der Turbulenz. 1. Mitteilung, Math. Phys. Klasse, Nachr.
Ges. Wiss. Göttingen, 21‒44 (Translated as NACA TM 609, 1931).
Schlichting H. (1933) Zur Entstehung der Turbulenz bei der Plattenströmung, Math. Phys. Klasse,
Nachr. Ges. Wiss. Göttingen, 181‒208.
Schubauer G. B. and Skramstad H. K. (1948) Laminar-boundary layer oscillations and transition
on a flat plate, NACA TN‒909.
Klebanoff P. S., Tidstrom K. D., and Sargent L. M. (1962) The three-dimensional nature of
boundary-layer instability, J. Fluid Mech., Vol. 12, pp. 1‒34.
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Schmid P.J. and Henningson D.S. (2000) Stability and transition in shear flows, Springer, p. 1-60.
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Litvinenko Yu.A., Grek G.R., Kozlov V.V. and Kozlov G.V. (2010)