Laminar to Turbulent Transition in Cylindrical Pipes

Course I: Fluid Mechanics & Energy Conversion

Laminar to Turbulent Transition in

Cylindrical Pipes

By,

Sai Sandeep Tallam IIT Roorkee

Mentors: Dr- Ing. Buelent Unsal Ms. Mina Nishi

Presentation Plan

z

Boundary Layer equations for laminar flow

z

Onset of turbulence

z

Primary stability theory

z

Simple examples on stability

z

Orr-Sommerfeld equation

z

Reyleigh criteria of stability

z

Origin of puffs and slugs

z

Natural and forced transition

z

Conclusion

Field Equations for flow

0 D divv Dt

ρ ρ

+ = r ( xx xy xz ) x Du p f Dt x x y z τ τ τ ρ = − ∂ + ∂ + ∂ + ∂ ∂ ∂ ∂ ∂ ( zx zy zz ) z Dw p f Dt z x y z τ τ τ ρ = − ∂ + ∂ + ∂ + ∂ ∂ ∂ ∂ ∂ Dv f gradP Div Dt ρ r = −r + τ ( yx yy yz ) y Dv p f Dt y x y z τ τ τ ρ = − ∂ + ∂ + ∂ + ∂ ∂ ∂ ∂ ∂ xx xy xz xy yy yz τ τ τ τ τ τ τ τ τ τ ⎛ ⎞ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟

The continuity Equation The Momentum Balance Equations

in all three coordinates

Vectorial representation of the momentum balance where

Boundary Layer Equations for laminar

flow

z Using the limit of Re to

infinity i.e. essentially invicid flows

z Continuity Equation

z Boundary condition where

derivatives of U wrt y vanish

z Putting the derivatives of U

in place of pressure gradient

z Considering the steady state

flow 2 2 1 u u v p u u v t x y ρ x ν y ∂ ∂ ∂ ∂ ∂ + + = − + ∂ ∂ ∂ ∂ ∂ 0 u v x y ∂ ₊ ∂ ₌ ∂ ∂ , ( , ) y → ∞ =u U x t 1 U U p U t x ρ x ∂ ₊ ∂ _{= −} ∂ ∂ ∂ ∂ 2 2 ( ) u u v U U u u v U t x y t x ν y ∂ ₊ ∂ ₊ ∂ _{= −} ∂ ₊ ∂ ₊ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 2 u v U u u v U x y x ν y ∂ ₊ ∂ _{= −} ∂ ₊ ∂ ∂ ∂ ∂ ∂

Plate Boundary layer and Blasius

equation

2 2 u v u u v x y ν y ∂ ₊ ∂ ₌ ∂ ∂ ∂ ∂ 0 u v x y ∂ ₊ ∂ ₌ ∂ ∂ ( ) u U_∞ =ϕ η ( ) y x η δ = d u dy ψ = d v dx ψ = − 2 xU f( ) ψ = ν _∞ η ' ( ) d u U f dy ψ _η ∞ = = ' ( ) 2 U d v f f dx x ν ψ _{∞ η} = − = − ''' ''

0

f

+ ×

f

f

=

Derivative of f wrt η η = 0: f=0 and df=0 η Æ α: df = 1

Onset of Turbulence

Transition in the boundary layer

z Boundary layer can also be either laminar or turbulent z The factors on which the transition depends are

– Re

– Pressure Distribution

– Nature of wall( Roughness) – Level of disturbance

z Waves namely Tollmien-Schlichting waves initiate the

transition from laminar to turbulent

z

Two dimensional Tollmien-Schlichting waves

are superimposed onto laminar boundary layer

at indifference Re.

z

This is Primary Stability Theory

z

Because of secondary instabilities three

dimensional and hence the

Λ

Structures

develop (secondary stability theory)

z

Λ

Vortices

are replaced by turbulent spots

completing the transition

Onset of Turbulence

Onset of Turbulence

z

Laminar turbulent transition is a

Stability

Problem

z

Small perturbations acted on the laminar flow

z

At small Re

– Damping action of viscosity large enough to dampen

these disturbances

z

At high Re

– Damping action not sufficient and hence disturbance

gets amplified and hence turbulence

Onset of Turbulence (Stability Theory)

Eigen value example

1 0 Re 2 0 Re x x y y − ⎛ ⎞ ⎜ ⎟ ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ₋ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ _{⎜ ⎟}⎝ ⎠ ⎝ ⎠ & & 2 1 0 1 and Re 1 1 Re ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ ⎝ ⎠ / Re 2 / Re ( ) 1 0 ( ) Re Re t t x t e e y t − − ⎛ ⎞ ⎛ ⎞ ₋⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

z Consider the system

z The eigen values of the matrix are negative z The eigen values are

Stability Example

z Consider

z Let the disturbance be of the form z On substitution and solving,

z The disturbance is damping with time

z The viscosity of the fluid eliminates the chance of

turbulence in the flow

z Gradients promoting turbulence

2 2 U U t y μ ρ ∂ ₌ ∂ ∂ ∂ 0 2 ( ) cos( ) U U t π y λ = 2 0 2 4 (0) exp U U γ π t λ ⎡ ⎤ = _⎢− _⎥ ⎣ ⎦

[

]

(0) exp ( ) U =U α β− t

z

Two methods of analysis

– Energy Method

– Method of small disturbances

z

Assuming 2 D disturbance and incompressible

flow

Î

2 D Navier Stokes Equation

z

Parallel Flow assumption

z

Superimposed values

z

Resulting motion

Onset of Turbulence (Stability Theory)

Primary Stability Theory

( ),

0; ( , )

U y V

=

W

=

P x y

' ' ' ( , , ), ( , , ), ( , , ) u x y t v x y t p x y t ' ' '

, , 0,

u

= +

U

u v

=

v w

=

p

= +

P

p

z

Inserting into Navier Stokes equation and

eliminating the quadratic terms

Onset of Turbulence (Stability Theory)

Primary Stability Theory

2 2 2 2 2 2 2 2 2 2 1 1 ( ) 1 1 ( ) 0 u u U P p U u u U v t x y x x y x y v v P p v v U t x y y x y u v x y ν ρ ρ ν ρ ρ ′ ′ ′ ′ ′ ∂ ∂ _′ ∂ ∂ ∂ ∂ ∂ ∂ + + + + = + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ′ ′ ′ ′ ′ ∂ ∂ ∂ ∂ ∂ ∂ + + + = + ∂ ∂ ∂ ∂ ∂ ∂ ′ ′ ∂ ∂ + = ∂ ∂

z Assuming that the basic flow itself satisfies the Navier

Stokes equations,

z On eliminating the pressure we have 2 equations and 2

unknowns

Onset of Turbulence (Stability Theory)

Primary Stability Theory

2 2 2 2 2 2 2 2 1 ( ) 1 ( ) 0 u u U p u u U v t x y x x y v v p v v U t x y x y u v x y

ν

ρ

ν

ρ

′ ′ ′ ′ ′ ∂ ₊ ∂ _{+ ′} ∂ ₊ ∂ ₌ ∂ ₊ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ′ ′ ′ ′ ′ ∂ ∂ ∂ ∂ ∂ + + = + ∂ ∂ ∂ ∂ ∂ ′ ′ ∂ ₊ ∂ ₌ ∂ ∂

z

Assume the following trial solution for the

stream function

z

α

is real and

β

is complex

z

Real part is the frequency and imaginary part

the amplification factor

z

the wave is damped else amplified

(unstable)

Onset of Turbulence (Stability Theory)

Orr-Sommerfeld Equation ( )

( , , )

x y t

( )

y e

i α βx t

ψ

₌

ϕ

− r i i

β β

= +

β

0 i β <

z Components of perturbation velocity

z Eliminating the pressure and 4th order terms,

.

z Inertial and frictional terms

z Starting point of stability in Laminar flows. This is called the

Orr-Sommerfeld Equation

Onset of Turbulence (Stability Theory)

Orr-Sommerfeld Equation ( ) ( ) i x t u y e y α β ψ ϕ − ∂ ′ = = ′ ∂ v′ = − ∂_∂ψ_x = −iα( )y ϕei(α βx− t) 2 2 4 ( )( ) ( 2 ) Re i U c ϕ α ϕ U ϕ ϕ α ϕ α ϕ α ′′ ′′ ′′′′ ′′ − − − = − − +

z

Of these the Re are generally fixed for a flow

and we vary the wave length

z

For a pair of this yields an eigen function

and eigen value

z

c

_i

is the curve of neutral stability and separates

the stable and unstable zone

Orr- Sommerfeld Equation

Parameters Re, ,

α

c

_r

and

c

_i

( )

y

ϕ

r i

Stable and Unstable zones

z

The point on the curve where the Re is smallest

z c_i = 0 is expected at high Re

z Neglecting the friction terms on the RHS of the OS

equation gives inviscid perturbation diff equation called Reyleigh Equation

z A second order differential equation with the boundary

conditions as follows

Reyleigh Equation

2

(

U

−

c

)(

ϕ α ϕ

′′

−

)

−

U

′′

ϕ

=

0

0; 0; : 0 y =

ϕ

= y = ∞

ϕ

=

z

Theorem 1

– The first important general statement of this kind is

the point of inflexion criterion. This states that the velocity profiles with point of inflection are unstable

z

Theorem 2

– A second important general statement says that in

the boundary layer profiles the velocity of

propagation for neutral perturbations (ci=0) is smaller than the maximum velocity of the mean flow

Reyleigh Theorem

r e

Controversy to Reyleigh Theorem 1

z

It is valid only in the case where the disturbance

amplified by a 2D wave is necessarily 2D

z

After break down of the T-S wave in the 2D

parallel flows the disturbance becomes 3D, a

type of spiral waves which proceed along the

stream direction

z

Hence the controversy

( )

( , , )

x y t

( , , )

y

e

i α βx t

ψ

_′

₌

ϕ

α β

−

Onset of Turbulence (Stability Theory)

On the eigenvalues of Orr-Sommerfeld Equation

z The solution along with the boundary condition defines a

characteristic equation

z Expanding the function about any point

Source: “On the eigenvaluesof the Orr-Sommerfeldequation”By,MGasterand R.Jordinson,MathematicsDepartment, University

( , ) 0 F α β = 0 0 (α β, ) 2 2 0 0 0 0 0 0 2 0 0 2 2 0 0 0 0 2 0 0 2 0 0 1 ( , ) ( , ) ( ) ( , ) ( ) ( , ) 2 1 ( ) ( , ) ( ) ( , ) 2 ( )( ) ... F F F F F F F α β α β α α α β α α α β α α β β α β β β α β β β α α β β α β ∂ ∂ = + − + − ∂ ∂ ∂ ∂ + − + − ∂ ∂ ∂ + − − + ∂ ∂

z Equating the above series to 0 and finding the values of

β for some known values of α

z Let α vary in a circle as

z The technique is to discretize the equation and then find

discrete values of the eigenvalues

z Other ways of finding out eigenvalues are

– Matrix iteration

– Shooting Technique

Involving direct numerical integration are solutions of algebraic equations

Onset of Turbulence (Stability Theory)

On the eigenvalues of Orr-Sommerfeld Equation

0 Re

iθ

Onset of Turbulence (Stability Theory)

Concluding Remarks

z Rapid calculation of eignvalues by representing β in

terms of α

z Simplification of derivatives as they are in turn

expressed as series

z Direct solution of Orr-Sommerfeld equation proved to be

tedious

z Contour integration of eigenvalues round a circle has

proven to be good choice for discretized behavior

Onset of Turbulence (Stability Theory)

z Orr Sommerfeld equation has an infinite set of discrete

eigenvalues and a corresponding complete set of eigenfunctions

Source: “The Continuous spectrum of the Orr-Sommerfeldequation. Part 1. The spectrum and the eigenvalues”,Journalof Fluid Mechanics (1978), vol. 87,part 1, pp. 33-54

Orr- Sommerfeld Equation

The Spectrum of eigenvalues

{ }

c

_n

z

Consider the wave equation

z

Solutions of the form

z

Here f(x) is of the form

z

Infinite set of discrete eigen values and eigen

vectors

Orr- Sommerfeld Equation

A Trivial Example

2 2 2 2 u u t x ∂ ₌ ∂ ∂ ∂ ( , ) ( ) i t u x t = f x eω 2 2 2 0 d f f dx +

ω

=

u

(0, )

t

=

u

(1, )

t

=

0

0.5 , ( ) 2 sin ( ) n n f xn n x

ω

₌

π

₌ −

π

z

These eigen values form a complete set

z

For an infinite domain,

z

If the second condition is relaxed then it forms a

continnum with real omega

z

Hence by this example they form a continnum

Orr- Sommerfeld Equation

A Trivial Example

(0, )

0, ( , )

0 as

u

t

=

u x t

→

x

→ ∞

0.5

( ; )

(2 )

sin

for

0

f x

ω

=

π

−

ω

x

ω

≥

z At Higher Re smooth and slightly disturbed inputs

transition occurs because of flow instabilities

z This causes Turbulent SLUGS and occupy entire cross

section

z At Re for (2000, 2700), for a disturbance at the inlet, the

turbulent regions carried forward i.e. convected downstream at a velocity slightly smaller than the average velocity result in structures called PUFFS

z Puffs occur at lower Re and Slugs at Higher Re.

Source: “On transition in a pipe. Part I. The Origin of puffs and slugsand the flow in a turbulent slug” By I. J WYGNANSKI AND F. H CHAMPAGNE,Journalof Fluid Mechanics, (1973), Vol59, part 2, pp. 281-335

z Slugs are caused by instability of

the boundary layer to small disturbances in the inlet region

– Associated with laminar to turbulent transition

– Observed for Re>3200

z Puffs, which are generated by large

disturbances at the inlet

– Incomplete relaminarization process

– Observed for 2000<Re<2700

z Slugs grow with axial distance and

merge leading to an increase in

intermittency factor and decrease in frequency

z No growth of puffs as ULE and UTE of puffs are

almost the same

z UTE decreases and ULE increases with Re

z This says ReÆlarge value UTEÆ0Îfully turbulent z For a fully turbulent flow, ULE = U and UTE = 0 and

slug is of same order of magnitude as the length of pipe

z These measurements are taken by keeping one fixed

and one moving turbulence detectors

Origin of Puffs and Slugs: Observations

z Irrespective of the type of disturbance, the flow conditions

inside a puff are the same

z Slugs do not originate at the entrance itself

z Slugs are product of transition in the developing boundary

layer downstream of the entrance

z Boundary layer spot begins on one side of the wall and

develops to comparable pipe size called the slug

z Breakdown of turbulence is a local phenomena and is not the

same across the cross section

z The final stage of slugs is that they breakdown into spots

Origin of Puffs and Slugs: Observations

z Velocity Profile interior of

the slug is some what similar to full turbulent flow

z Normalized fluctuation

velocity plotted against normalized radial velocity

z Observed that the level

of fluctuations inside a slug are greater than that in fully turbulent flow

Origin of Puffs and Slugs: Observations

The Equilibrium Puff

z

Puffs are generated by large disturbances at inlet

z

All puffs at same Re are equal in length

z

Turbulent activity if puff is strongest in central zone

z

No distinction between turbulent and non turbulent

zone at the leading end of puff while that is not the

case with turbulent slug

z

Fluid might enter and leave the puff from the same

interface

Source: “On transition in a pipe. Part II. The Equilibrium Puff”, By I. J WYGNANSKI,M SOKOLOV AND D.FRIEDMAN,Journalof Fluid Mechanics, (1975), Vol69, part 2

Natural Transition

Pressure Drop and Slug

2 2 w f U τ ρ = 64 Re lam f = 0.25 0.3614 / Re turb f =

z Slug starts at entrance and reaches the exit

z Slug tail moves through the pipe and finally leaves

2 2 _w f U τ ρ = 64 Re lam f =

Forced Transition

z To cause slugs to occur at low

Re obstacles could be

introduced at wall and pipe inlet

z There exists a range of Re

which is critical depending on the height of obstacle

z Transition via puffs at lower

Conclusions

z Tollmien-Schlichting waves initiate the laminar turbulent

transition

z Stability analysis is helpful to find the indifference Re z Formation of puffs and slugs and then to the turbulent

spots and finally the fully developed turbulent flow

z Puffs occur at lower reynolds number than the slugs

z Turbulence activity inside a slug is irrespective of source z At same Re all Puffs have same length

z The boundaries of the slugs are relatively clearly

defined than that of the puffs

z For increasing Re the slug dimensions increase