ae 549 linear stability theory and laminar-turbulent ...ae549/introduction.pdf · ae 549 linear...

AE 549 Linear Stability Theory and

Laminar-Turbulent Transition

Prof. Dr. Serkan ÖZGEN Introduction

Historical Background

Early attempts to understand transition did not care about

turbulence or the details of its initial appearance but tried to

explain why the original laminar flow can not exist indefinetely.

Rayleigh (1880-1913) produced results concerning the

instability of inviscid flows but little progress was made

towards the original goal. The inflectional instability was

discovered by Rayleigh.

Taylor (1915) and Prandtl (1921) indicated that viscosity can

destabilize a flow.

Tollmien (1929) and Schlichting (1935) outlined a complete

theory of boundary-layer stability and calculated the total

amplification of the most unstable frequencies.

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Historical Background-Reynolds experiments

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Historical Background

Schubauer and Skramstad (1947) performed experiments, which demonstrated the presence of instability waves in a boundary-layer, their connection with transition and the quantitative description of their behavior by the theory of Tollmien and Schlichting.

Smith and Gamberoni (1956) and van Ingen (1956) devised a transition prediction method (en method), which is still widely used today.

Pretsch (1942) provided a large body of numerical results by calculating the stability characteristics of Falkner-Skanvelocity profiles.

In the 1960s, the advent of digital computers permitted solutions to be achieved for many boundary-layer flows like 3-D boundary-layers, compressible boundary-layers, unsteady boundary-layers, etc.

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Practical Aspects

The problem of understanding the origins of turbulent flow

and transition to turbulent flow are the most important

unsolved problems of fluid mechanics and aerodynamics.

There is a large number of applications for information

regarding transition location and the details of the

subsequent turbulent flow.

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Practical Aspects

A few examples:

Nose cone and heat shield requirements on reentry vehicles are

critical functions of transition altitude.

If transition can be delayed with laminar flow control on the wings of

large transport aircraft, substantial savings in fuel will result.

Efforts to accurately predict airfoil surface heat transfer and devise a

cooling mechanism for turbine blades are not fully effective due to

absence of a fully reliable transition prediction technique.

Separation and stall on low-Reynolds number airfoils and turbine

blades strongly depend on whether the boundary-layer is laminar,

transitional or turbulent.

The performance and detection of submarines and torpedoes are

influenced by turbulent flows and efforts directed towards drag

reduction require details of the transition process.

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Practical Aspects

Because of the presence of different factors like disturbance

environment, surface geometry, roughness, heat

transfer, noise level, etc. on transition, it is not possible to

develop general prediction schemes for transition location.

As can be anticipated, no mathematical model exists that can

predict the transition location for a given flow.

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Basic Concepts of

Hydrodynamic Stability Theory

In order to analyze the stability problem of a laminar flow, one needs to know the velocity, pressure and temperature fields at any point , and time, t, which define the basic flow. The basic flow may be steady or unsteady, but must satisfy the corresponding equations of motion and boundary conditions.

In very simple words, instability occurs because there is a disturbance of the equilibrium of the forces acting on the system, namely external, inertial and viscous forces.

External forces are buoyancy in a fluid of variable density, surface tension, magneto-hydrodynamic forces, centrifugal and coriolisforces.

If a small disturbance is introduced to the flow, it may die out, may preserve its initial amplitude, or it may amplify. Such disturbances are called as stable, neutrally (or marginally) stable or unstable disturbances, respectively.

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x

Basic Concepts of


Roles of different factors in the stability mechanism:

Gravity: destabilizing if heavier fluid lies on top of lighter fluid,

stabilizing if vice-versa.

Surface tension: tries to decrease the area of a contact surface,

therefore stabilizing.

Viscosity: viscosity has a dual role in the instability mechanism. All

other effects being absent, the onset of instability in a boundary-layer

flow is determined by the relative magnitudes of these two effects.

Viscosity is stabilizing because it dissipates the energy of the

disturbances (low Reynolds numbers).

Viscosity is destabilizing because it diffuses momentum (high

Reynolds numbers).

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Basic Concepts of



Surface curvature: both concave and convex surfaces enhance

instability but instability in concave surfaces is stronger and leads to

Görtler vortices.

For a convex surface, the instability can diffuse into the freestream

but in a concave surface it remains in the boundary-layer and is

convected downstream.

Pressure gradient: a favorable pressure gradient accelerates

the basic flow so the kinetic energy increases, which is a

stabilizing effect.

On the other hand, an adverse pressure gradient decelerates the

flow, so is destabilizing.

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Basic Concepts of



Thermal conductivity, convection: these factors have similar

effects as viscosity. These factors in general smooth out

temperature differences, so are stabilizing.

Boundaries of the flow: this is probably the most important factor

influencing the instability mechanism.

Boundaries constrain the development of a disturbance. Flow is

more stable when boundaries are close together.

Strong shear generated by the walls is diffused out by viscosity,

which is a destabilizing effect.

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Basic Concepts of



Velocity profile shapes: profiles with an inflection point are

unstable. This is called Rayleigh’s inflection point theorem.

The location where the wave speed equals the basic flow velocity is

called the critical layer.

It has been shown to play an important role in transforming the

energy from the basic flow to the disturbances.

Maximum axial disturbance velocity occurs near the critical layer (for

a flat plate boundary-layer wave speed ≈ 0.3Ue).

In a typical flow, one or more of these mechanisms may act.

In a flat plate flow boundary-layer, both roles of viscosity,

inertia and boundaries contribute to the stability problem.

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Basic Concepts of


When viscous dissipation > viscous diffusion, the flow is stable

(low Re).

When Reynolds number increases, diffusion of momentum

from thin shear layers near the wall lead to instability.

Critical Reynolds number: represents the balance between

the destabilizing shear forces and stabilizing viscous forces.

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Basic Concepts of


Perturbations leading to instability may arise from small changes in the boundary conditions due to surface roughness, freestream turbulence, noise, etc. These constitute the disturbance environment. How these disturbances are entrained into the flow is a subject of receptivity.

Receptivity: mechanisms that cause disturbances to enter the flow and create initial amplitudes for unstable waves.

This is not a very well understood process, however, it provides initial conditions of amplitude, phase and frequency for the instability waves that lead to turbulence.

These disturbances may be so small that they may not be measurable with available instruments and can only be observed after the onset of instability.

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Basic Concepts of


A variety of instabilities can occur independently or together.

The initial growth of these disturbances is defined by the

linear stability theory.

As the amplitudes of the disturbances grow, three-dimensional

and/or non-linear interactions occur and linear stability theory

can no longer be used. Eventually, breakdown to turbulence

occurs.

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Stages of Laminar-Turbulent Transition

in a Boundary-Layer

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in a Boundary-Layer

1. Stable, laminar flow following the leading edge,

2. Unstable, laminar flow with 2-D Tollmien-Schlichting waves,

3. Development of unstable, laminar, 3-D waves and vortex

formation,

4. Burst of turbulence in places of very high local vorticity,

5. Formation of turbulent spots in places where the turbulent

velocity fluctuations are large,

6. Coalescence of turbulent spots into a fully developed

turbulent boundary layer.

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in a Boundary-Layer

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in a Boundary-Layer

By-pass mechanisms can cause the transition sequence to

by-pass one or more of these stages.

Three of the natural by-pass mechanisms:

1. Görtler vortices,

2. Surface roughness,

3. Freestream turbulence and noise.

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in a Boundary-Layer

Görtler vortices occur in concave surfaces.

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in a Boundary-Layer

The centrigugal force acts in the direction of smaller velocity.

This is like heavier fluid flowing over a lighter fluid (like a

water-oil flow).

As a result, the particles subject to greater centrifugal forces

will penetrate into the particles subject to smaller centrifugal

forces.

As a result, peak and valley structures called Görtler vortices

will occur even before 2-D Tollmien-Schlichting waves and

3-D instabilities will be observed.

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in a Boundary-Layer

Surface roughness is known to effect transition because of

the disturbances introduced by its presence.

This is a wedge shaped region of turbulent flow originating at

the roughness element and extending downstream.

If we consider an isolated roughness element on the surface,

depending on its shape, external velocity distribution,

location on the body and the flow velocity, there is a critical

roughness height, kcrit , below which the roughness

element has no effect on transition, i.e. The transition

location is the same as that on a smooth surface.

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in a Boundary-Layer

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in a Boundary-Layer

As the height of the roughness element increases, a second

critical value, k*crit is reached for which the transition occurs

at the immediate downstream of the roughness element.

This is a wedge shaped region of turbulent flow trailing

downstream with the roughness element at its apex and with

a semi-angle ≈10o.

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in a Boundary-Layer

Intensity of freestream turbulence:

Transition Reynolds number increases with decreasing

freestream turbulence intensity until a maximum is reached.

Further reductions in turbulence intensity do not improve or

delay transition.

Corrections to smooth transition data, e.g. to the en data.

n = -8.43 - 2.4Tu

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∞

2

u U

u′=T


in a Boundary-Layer

In addition to natural by-pass mechanisms, there are artificial

by-pass mechanisms which are used as research tools or

means of control:

Vibrating ribbon (maybe spanwise periodic as well),

Pneumatic turbulators.

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Receptivity

Amplitude and spectral (frequency) characteristics of the

disturbances inside the laminar boundary-layer strongly

influence which type of transition occurs and the major need

in this area is to understand how freestream disturbances are

entrained into the boundary-layer, namely the receptivity

problem.

Receptivity: mechanisms that cause freestream

disturbances to enter the boundary-layer and create initial

amplitudes for unstable waves.

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Receptivity

If initial amplitudes of disturbances are small, linear

modes of the boundary-layer will be excited, which will

result in the Tollmien-Schlichting waves.

If initial amplitudes are large non-linear modes will be excited

and premature transition will occur through the by-pass

mechanisms.

Mathematically, the receptivity problem is also different from

the stability problem.

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Receptivity

Receptivity has many different paths to introduce a

disturbance into the boundary layer. These include:

Interaction of freestream turbulence and acoustical

disturbances,

Leading edge curvature,

Discontinuities on the surface contour or inhomogenuities.

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Receptivity

Stability problem deals with normal mode disturbances

within the boundary-layer. These are determined from the

solution of linearized Navier-Stokes equations and are

ordinary differential equations and the problem becomes an

eigenvalue problem.

On the other hand, in the receptivity problem, neither the

equations nor the boundary conditions are homogeneous.

Boundary-layer is forced externally by a disturbance. Thus,

the receptivity problem is an initial value problem and

requires the solution of full Navier-Stokes equations.

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Elements of Stability Theory

The stability theory is concerned with individual sine

waves propagating in the boundary-layer, parallel to the

wall.

Amplitudes of the waves vary through the boundary-layer

and are small enough so that linear theory may be used.

Frequency of a wave is and the wave numbers is

k = 2, where is the wavelength.

Two-dimensional waves: lines of constant phase normal to

the freestream direction.

Oblique waves: wavenumber is a vector.

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Phase velocity, c < freestream velocity U

At some point in the boundary-layer, the mean flow velocity

is = c (critical layer).

Wave amplitude usually has a maximum near the critical

layer.

Numerical results calculated from stability theory are usually

presented in a Re- space.

Recr: the Reynolds number below which no amplification is

possible. It only tells where the instability starts. Recr ≠Retr .

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A wave which is introduced into a steady boundary-layer

with a particular frequency will preserve its frequency as it

propagates downstream, while the wavenumber will change.

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A wave which is introduced into a steady boundary-layer

with a particular frequency will preserve its frequency as it

propagates downstream, while the wavenumber will change.

0 Reo: damping,

Reo Re1 : amplification,

Re1 : damping.

If the amplitude of a wave becomes large enough before Re1

is reached, then the nonlinear processes which eventually

lead to transition will take over, and the wave will continue to

grow even though the linear theory says it should damp.

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The linear stability theory can be used to find:

Amplification and damping rates,

Frequency, wavenumber and Reynolds number of waves,

Amplification rates as a function of frequency at a given Re,

Amplitude history of a constant frequency wave as it travels through

the unstable region.

This can be calculated as a ratio of the amplitude to some (generally

unknown) initial amplitude once the amplification rates are known.

Given some initial disturbance spectrum, it is possible to identify the

frequency whose amplitude has increased the most at each Reynolds

number. It is probably one of these frequencies that triggers the

whole transition process after reaching a critical amplitude.

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