Momentum Heat Mass Transfer

Momentum Heat Mass TransferMHMT7

Turbulent flow. Statistical characteristics, kinetic energy of fluctuations. Transition laminar/turbulence. Velocity and stress profiles in self-preserving flows and at walls. Time averaging of fluctuations applied to NS equations. Boussineq concept of turbulent viscosity. Turbulent models (Prandtl, k).

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

Turbulent flows

sourceDtD

NS equations and TurbulenceMHMT7

The convection term is a quadratic function of velocities – source of

nonlinearities and turbulent phenomena

2( ( ))u uu p u ft

2

( )Re uu uDu

Relative magnitude of inertial (nonlinear) and viscous (linear) terms is Reynolds number

Navier Stokes equations are nonlinear even in the case that the viscosity is constant. Primary source of nonlinearities in the Navier Stokes equations are convective inertial terms

TurbulenceMHMT7

Increasing Re increases nonlinearity of NS equations. This nonlinearity leads to sensitivity of NS solution to flow disturbances.

Laminar flow: Re<Recrit

Turbulent flow: Re>Recrit

Distarbunce is attenuated

Distarbunce is amplified

Velocity field (controlled by NS equations) depends on boundary and initial conditions. Let us assume steady boundary conditions (time independent velocity/pressure profiles at inlets/outlets): There will be only one (or two) steady solutions in the laminar flow regime, independent of initial conditions (starting velocity field). On the other hand the flow field will be unsteady and depending upon initial conditions (sometimes will be sensitive even to infinitely small disturbances) in the turbulent flow regime.

Turbulence - fluctuationsMHMT7

Turbulence can be defined as a deterministic chaos. Velocity and pressure fields are NON-STATIONARY (du/dt is nonzero) even if flowrate and boundary conditions are constant. Trajectory of individual particles are extremely sensitive to initial conditions (even the particles that are very close at some moment diverge apart during time evolution).

Velocities, pressures, temperatures… are still solutions of NS and energy equations, however they are non-stationary and form chaotically oscillating turbulent vortices (eddies). Time and spatial profiles of transported properties are characterized by fluctuations

'u u u Actual value

at given time and

space

Mean value

fluctuation

Statistics of turbulent fluctuations

Mean values (remark: mean values of fluctuations are zero)

rms (root mean square)

Kinetic energy of turbulence

Intensity of turbulence

Reynolds stresses

, , ,u p T

2'u

2 2 21 ( ' ' ' )2

k u v w

2( ) /3

I k u

2 2 2, , , , ,' , ' , ' , ' ', ' 't xx t yy t zz t xy t xzu v w u v u w

Turbulence fluctuationsMHMT7

E()

=2f/u1/L 1/

Large energetic eddies (size L) break to smaller eddies. This transformation

is not affected by viscosity

Inertial subrange (inertial effects dominate and spectral energy

depends only upon wavenumber and )

Smallest eddies (size is called Kolmogorov scale) disappear, because kinetic energy is converted to heat by

friction

Spectral energy of turbulent eddies

wavenumber (1/size of eddy)

Kinetic energy of turbulent fluctuations is the sum of energies of turbulent eddies of different sizes

/ /

0

1 ( )2 i ik u u E d

Typical values of frequency f~10 kHz, Kolmogorov scale ~0.01 up to 0.1 mm

Kolmogorov scale decreases with the increasing Re

Turbulent eddies - scales MHMT7

Kolmogorov scales (the smallest turbulent eddies) follow from dimensional analysis, assuming that everything depends only upon the kinematic viscosity and upon the rate of energy supply in the energetic cascade (only for small isotropic eddies, of course)

34

4u

Length scale Time scale velocity scale

These expressions follow from dimension of viscosity [m2/s] and the rate of energy dissipation [m2/s3]

2 2 2 2

3 3[ ] [ ][ ]

1 2 20 3

m m mms s s

3 / 41/ 4

Turbulent eddies - scales MHMT7

Transition laminar-turbulentMHMT7

Hopper

There always exists a steady solution of NS equation corresponding to the laminar flow regime, but need not be the only one. Sometimes even an infinitely small disturbance is enough for a jump to unsteady turbulent regime, sometimes the disturbance must by sufficiently strong (typical for viscous instabilities). Therefore the transition from laminar to turbulent flow regime cannot be always specified by a unique Recrit because it depends upon the level of disturbances (laminar flow can be achieved in pipe at Re10000 in laboratory).

velocity

y

Hydrodynamic instabilities are classified as

Inviscid instabilities

characterised by existence of inflection point of velocity profile

- jets

- wakes

- boundary layers wit adverse pressure gradient p>0

Viscous instabilities

Linear eigenvalues analysis (Orr-Sommerfeld equations)

- channels, simple shear flows (pipes)

- boundary layers with p>0 Poiseuille flow ~ 5700

Couette flow – stable?

velocity

y Inflection – source of instability

(max.gradient)

There is no inflection of velocity profile in a pipe, however turbulent regime

exists if Re>2100


How to indentify whether the flow is laminar or turbulent ?

Experimentally

Visualization, hot wire anemometers, LDA (Laser Doppler Anemometry).

Numerical experimentsStart numerical simulation selected to unsteady laminar flow. As soon as the solution converges to steady solution for sufficiently fine grid the flow regime is probably laminar

Recrit

According to the value of Reynolds number using literature data of critical Reynolds number



Geometry Recrit

Jets 5-10

Baffled channels

100

Couette flow

300

Cross flow 400

Planar channel

1000

Geometry Recrit

Circ.pipe 2000

Coiled pipe 5000

Suspension in pipe

7000

Cavity 8000

Plate 500000

D

D

D/2

D

D

D

D

D

Jets and mixing layerMHMT7

Free flows (self preserving flows)

Jets Mixing layers Wakes

h yu

umax

umax

umin

uyb

max

( )u yfu h

min

max min

( )u u ygu u b

max

max

constux

constux

Circular jet

Planar jet

x

x

Jet thickness ~x, mixing length ~x see Goertler, Abramovic Teorieja turbulentnych struj, Moskva 1984:

Turbulence at wallsMHMT7

Flow at walls (boundary layers)y

Laminar sublayer

Buffer layer

Log law * wu

*

* u yuu yu

Friction velocity

laminar y+<5 u+=y+

buffer 5<y+<30 u+=-3+5lny+

Log law 30<y+<500 u+=lnEy+/

INNER layer (independent of bulk velocities)

OUTER layer (law of wake) max

*

1 ln( )u u y Au

u

Turbulence at wallsMHMT7

Flow at walls (turbulent stresses)t

*yuy

2 2( )

[1 exp( )]26

t m

m

uly

yl y

2 2' ' 24 [1 exp( )]26t wyu v

Prandtl’s model

vanDriest model of mixing length lm 5 30

Time averaging fluctuationMHMT7


RANS (Reynolds Averaging of Navier Stokes eqs.)

1 ( ) '

( ) ''

' '

t t

t

t t

tt t

t

t dtt

t dt

dt

Time average

Favre’s average

Proof: ( ')( ') ' '

Remark: Favre’s average differs only for compressible substances and at high Mach number flows. Ma>1


0 0 0u u u

( ) ( ) ( )

( ) ( )) ) ( ' '(

u uu p ut

u ut

u uu p u

Continuity equation

Navier Stokes equations

Reynolds stresses

' 't u u


( ) ( ) ( )

( ) ( ') ( ')) ( u

ut

ut

Turbulent fluxes

' 'tq u

Fourier Kirchhoff, mass transport, kinetic energy equations for transport of scalars

Turbulent viscosityMHMT7

It was Boussinesq who suggested: Why not to calculate the turbulent Reynolds stresses and turbulent heat fluxes in the same way like in laminar flows, just replacing actual viscosity by an artificially increased turbulent viscosity and to replace actual driving potential (rate of deformation) by driving potential evaluated from averaged velocities, temperatures,…

…and this is Joseph Boussinesq

this is self portrait of Hopper

Turbulent viscosityMHMT7

Rate of deformation based upon gradient

of averaged velocities

' ' tu

' 'i ti

ux

' ' ( ( ) )Ttu u u u

1 ( ( ) )2

TE u u 2t tE

' ' ( )j ii j t

i j

u uu u

x x

Analogy to Fourier law

Analogy to Newton’s law

q T

( ( ) )Tu u

Turbulent viscosity - modelsMHMT7

Prandtl’s model of mixing length. Turbulent viscosity is derived from analogy with gases, based upon transport of momentum by molecules (kinetic theory of gases). Turbulent eddy (driven by main flow) represents a molecule, and mean path between collisions of molecules is substituted by mixing length.

Excellent model for jets, wakes, boundary layer flows. Disadvantage: it fails in recirculating flows (or in flows where transport of eddies is very important).

2t m

uly

Mixing length

h

h

y

u(y) 0.075ml h0.09ml h

0.07ml h

[1 exp( )]26myl y

Circular jet

Planar jet

Mixing layers

Currently used algebraic models are Baldwin Lomax, and Cebecci Smith


Two equation models calculate turbulent viscosity from the pairs of turbulent characteristics k-, or k-, or k-l (Rotta 1986)

k (kinetic energy of turbulent fluctuations) [m2/s2]

(dissipation of kinetic energy) [m2/s3]

(specific dissipation energy) [1/s]

2

t

t

k

kC

Wilcox (1998) k-omega model, Kolmogorov (1942)

Jones,Launder (1972), Launder Spalding (1974) k-epsilon model

3 /u l

C = 0.09 (Fluent-default)

].1

[2

2

3 smkg

s

sm

mkg

4

4

23

3

[ ].

mkg kgsmm m ss


1 2( ) ( ) ( 2 : )ttu C E E C

t k

Standard k- modelTransport equation for kinetic energy of turbulence (see also lecture 9)

convective transport diffusive transport by fluctuating turbulent eddies

production term (energy of main flow converted to energy of turbulent eddies)

dissipation kinetic energy of turbulence to heat

Similar transport equation for dissipation of kinetic energy

this term follows from inspection analysis

]1[

2

2

3

2

ssmsm

k

( ) ( ) 2 :tt

k

k ku k E Et

CFD modelling - FLUENTMHMT7

This is example of 2 pages in Fluent’s manual (Fluent is the most frequently used program for Computer Fluid Dynamics modelling)

EXAMMHMT7

Turbulence

What is important (at least for exam)MHMT7

Statistical characteristics (what is it the second correlation?)

2 2 21 ( ' ' ' )2

k u v w

2( ) /3

I k u

2 2 2, , , , ,' , ' , ' , ' ', ' 't xx t yy t zz t xy t xzu v w u v u w

Kinetic energy

Intensity of turbulence

Reynolds stresses

Definition of turbulence – deterministic chaos

(stability of solution of NS equations, explain role of convective acceleration)


Re uD

Transition laminar-turbulent flows. Critical Reynolds number

D D

DD

10

400

2100

500000


* wu

*

* u yuu yu

Velocity profile at wall u+(y+)

Friction velocity

Laminar sublayer y+<5

(how to calculate w?)


Boussinesq hypotheses of turbulent viscosity

1 ( ( ) )2

TE u u

2t tE

2t m

uly

Prandtl’s model of turbulent viscosity

(what is the difference between and ?)E

(what is it lm?)

Principles of k-epsilon models. Derive turbulent viscosity as a function of k and .

Momentum Heat Mass Transfer

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