july 2004singapore turbulence colloquium turbulence modelling of buoyancy-affected flows brian...

July 2004 Singapore Turbulence Colloquium

Turbulence Modelling of Buoyancy-Affected Flows

Brian LaunderUMIST, Manchester, UK


2

Aims of Lecture

To give an impression of what level of RANS modelling is needed to predict different types of gravity-affected shear flow phenomena

To show some illustrative predictions of buoyant and stratified flow ______________________________ Even if buoyant/stratified flows are of no interest,

much of the lecture has parallels with other agencies modifying turbulence, e.g. system rotation


3

An important distinction

A buoyant flow is one where the main effects of the gravitational vector are on the mean motion.

A stratified flow is one where the main effects of the g vector are on the fluctuating motion and thus, after Rey-nolds averaging, on the 2nd moments

Vertical flows are usually “buoyant”; ….horizontal flows usually “stratified”.


4

A modelling consequence

Buoyant flows can often be computed with simple eddy viscosity models… at least close to a wall.

Stratified flows always require at least some form of 2nd-moment closure (perhaps truncated) and maybe even 3rd-moment or U-RANS modelling to capture the flow behaviour sufficiently closely


5

Refinements to k-ε EVM for (mainly) vertical, buoyant flows near walls

In many simple flows NO refinement needed because shear generation of k (and ε) much larger than buoyant generation.

Note buoyant k-generation is: <ρ’ui>gi/ρ

or G = - β<uiθ>gi; β Vol. exp’n coeff

If x2 is vertically up: gi = [0, -g, 0]; Θ = Θ(x1); U=U(x1)

G = + βg<u2θ> = 0 with SGD but … = - cθcμβ(k3/ε2)Θ/x1U2/x1 with GGDH

ie.horizontal temp. gradients cause buoyant effects.

This refinement brings (usually minor) improve-ments to vertical flows (e.g. Ince & Launder, 1989)


6

A problem with flow near walls If source of buoyancy arises through heat

transfer through a wall, the wall sub-layer region is principally affected.

Flow is then far from local equilibrium But equally, for an industrial flow simulation,

one needs to avoid using a fine-grid “low-Reynolds-number” model.

Two new wall-function approaches have been developed to remove these problems: PhD’s and papers by Gerasimov and Gant


7

Gerasimov’s analytical wall functions – a recap

Based on a simplified analytical solution of wall-parallel momentum equation assuming a linear eddy viscosity variation over wall cell.

Gravitational term appears in momentum equation and thus affects the form of the wall function.

Vital to include the effects of thickening and thinning of the viscous sublayer.

Reduces computing budget by at least by factor 10 compared the ‘low-Reynolds-number’ treatments.


8

Application to up-flow in a heated pipe with k-ε EVM; Gerasimov(2003)

50 100 150x/d

0

20

40

60

80

100

Nu

Exp.data of Li (1994)y*

n=50yn

*=75y*

n=100yn

*=150Nu=0.023 Pr0.333 Re0.8

LRN Calculation

Inlet: Re=15023, Gr=2.163*108, Bo=0.1124

(a)

LRN terms included


9

Downward flow in an annulus with core tube heated; Gerasimov

(2003)

0 10 20 30 40 50 60 70 80x / deff

0

20

40

60

80

100

120

140N

uExperimentAWFStWFLRN (buoyant case)LRN (forced convection)

Annular Flow: Re=4000, Bo=2.89

(a)

0 10 20 30 40 50 60 70 80x / deff

290

295

300

305

310

315

320

325

330

335

340

Tw

all,K

(b)


10

How to extend linear EVM’s to stratified flows?

When a linear EVM is applied to horizontal or recirc-ulating flows, the model is often made sensitive to gravitational influences through the ε-equation.

Typically: Gravitational Source = Cε3Gε/k

Different values taken for Cε3 according to whether the flow is horizontal or vertical; stable or unstable…

Others have made Cε3 ~ tanh (vvert/vhoriz) [Fluent]

These approaches are NOT recommended unless extensive calibration against experiments similar to those to be predicted has been made!


11

Some guidelines in computing stratified flows

Don’t use a linear EVM. Incorporate primary gravitational effects

through constitutive equations for stresses and heat flux

Give the shear and gravitational terms equal weight in the ε equation … unless you are considering just a single flow type and are confident to tune


12

Where does buoyancy enter the turbulence equations?

In the second-moment equations (by way of the Navier Stokes equations):

<φDui /Dt> = ….<φρ'>gi /ρ

(where ui is turbulent velocity and denotes uj , θ, etc)… and in the pressure-strain term (see later) In the length scale or ε equation?? Our recommended practice is to handle G identically to

P in the scale equation… though this matter is not entirely settled.

13 Singapore Turbulence Colloquium

July 2004

Second moment equations with buoyant terms

Du u

DtP G d

i jij ij ij ij ij

G g u g uij i j j i

Du

DtP G di

i i i i i

G gi i 2

D

DtP d

2


14

Why a simple ε-equation fix doesn’t work

Consider a simple shear flow with x1 the flow direction and x2 that of velocity/temp gradient.

If x1 is horizontal, x2 vertical the g vector acts as in upper diagram

If x1 is vertical, x2 horiz-ontal, the lower figure shows the g impacts.


15

Features of the transport equations

‘Invisible’ gravitational effects also act on the pressure fluctuations

Possible effects also in the ε equation Note <θ2> contains no pressure fluctuations

…but εθθ hard to model. Usually we assume thermal time scale <θ2>/εθθ is linked to the dynamic time scale… though somewhat affected by anisotropy of the heat flux


16

Buoyant effects on pressure correlations: 1 –Basic Model

The IP model for shear assumes:

ij2 = -c2 [Pij - ⅔ijP]

with c2 0.6 (exact for isotropic turb.) By analogy we take: ij3 = -c3 [Gij - ⅔ijG] Now c3 = 0.3 in isotropic turbulence but suggest optimum value in the range 0.5 0.6. Pressure fluctuations in heat flux equations: i = - c3Gi

Isotropic turbulence suggests c3 = ⅓ but it is usually taken equal to c3.

Wall-reflection effects must be added but calibration has been done only for an infinite plane wall.


17

Buoyant effects on pressure correlations: 2 – TCL Model

Application of two-component-limit principles to buoyant parts of pressure fluctuations produces forms with NO empirical constants.

The TCL form of ij3 is very long (see Craft & Launder, 2002 in “Closure Strategies” book, p. 411)

i3 = ⅓βgi<2> - βgkaik

Note both ij3 and i3 satisfy isotropic and 2-component limits.

NO wall corrections beyond viscosity affected sublayer


18

Handling the dissipation rates Standard equation for but with c3 = c1

For TCL version c1 = 1.0 and c2 = 1.92/(1.+0.7AA2½)

where A2 aijaji , A3 aijajkaki and A 1 - (9/8)[A2 – A3]

Local isotropy and “proportional timescales” for scalars and their fluxes

kc

k2

)GP(cd

Dt

D 2

2kkkk

1

k

R;0

2

i

k/uuA;A12

3R 2

ii22


19

Modelling the transport terms

Diffusive transport usually handled by GGDH:

<uk> = - c<ujuk>(k/)<>/xj

…or it may be simplified to give an algebraic set of equations, e.g Rodi (1971):

conv.- diff’n of <> = [conv. - diff’n of k]×{<>/k}

= [P - ] <>/k

But situations also arise where transport is very important and then 3rd-moment closure may be necessary.


July 2004

Comparison of spreading rates for plumes


21

Application 1: Downwards directed warm jet into cool

“pool” In practice “pool” moves

slowly vertically up to give a steady flow.

Profiles of mean velocity (c) and shear stress (b) better represented by TCL scheme (solid line) than basic model (broken line).

From Cresswell et al (1989)


22

Applications 2: 3D surface jet

Not a stratified flow but clearly related.

Note Basic Model shows too strong asymmetry in spreading rates; TCL scheme approximately correct

From Craft, Kidger & Launder (2000)


23

Applications 3: The downwards directed buoyant

wall jet Downward directed

warm wall jet collides with slow-moving up-flow which causes jet to turn around.

Buoyant effects important.

LES data of Laurence et al(2003) extend exp’ts by Prof. Jackson’s team.

Outflow region

Jet

x

y

Cold upflow


24

Penetration depth with k- and TCL models versus LES for

isothermal flow

0 0.2 x0

0.2

0.4

0.6

0.8

1

1.2

1.4

Y

Standard WFTCL model

(g)0 0.2 x

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Y

Analytical WFBasic 2nd MC

(f)0 0.2 0.4 0.6 x

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Y

Level V15 0.06514 0.04913 0.03312 0.01811 0.00210 -0.0149 -0.0308 -0.0457 -0.0616 -0.0775 -0.0934 -0.1093 -0.1242 -0.1401 -0.156

Analytical WFTCL model

(h)0 0.2 x

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Y

LRN Calculationk- model

(b)0 0.2 x

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Y

(a)

LES


25

Turbulence energy using 2nd moment closure – buoyant case Basic 2nd

moment closure no better than k-ε EVM

TCL model mimics turbulence energy levels much better


26

k- prediction of temperature field Too little mixing

predicted by all forms so high temperatures (red) remain longer.

Again “standard” WF’s far different from “low-Re” result even though no heat transfer through the wall


27

2nd moment closure predictions of thermal field

“Basic model shows too little mixing and too deep penetration with St. WF

TCL-AWF gives most rapid mixing, closest to experiment


28

Application 4:Stably stratified mixing layer

NB: k- EVM creates far too much mixing but the two 2nd moment closures exhibit unphysical over- and under-shoots due to inadequate diffusion model


July 2004

Stratified mixing layer -2

Kidger (1999) developed 3rd moment closure for components containing the density fluctuations, e.g.

P1, P2, G: generation by 1st, 2nd moments and gravitational effects. pressure interactions, etc.

For further details of modelling see Launder & Sandham (2002) pp 417 - 418

Du

DtP P G dk

k k k k k k 2

1 2

Du u

DtP P G dk j

kj kj kj kj kj jk

1 2


30

Stratified mixing layer - 3:TCL 2nd and 3rd moment closures compared


31

Conclusions -1

Different levels of modelling are needed for buoyant and stratified flows.

Low-Re k- modelling successful for buoyant up- and down-flow in pipes and channels (but make sure Ret k2/ not yk½/)

Gerasimov’s analytical wall functions perform well in such flows, saving typically 90% of the computer time compared with low-Re model.

Buoyantly modified vertical free shear flows (plumes) not well modelled at this level


32

Conclusions -2

The TCL model mimics observed behaviour in stratified flows (consistent with non-buoyant results).

No new empirical coefficients introduced and no wall-reflection requirements.

Treat P and G identically in equation. In strongly stratified flows one may need to go to

3rd-moment closure for triple moments containing density fluctuations.

Unsteady RANS approach also powerful for stratified flows – see lecture by Prof Hanjalić.

july 2004singapore turbulence colloquium turbulence modelling of buoyancy-affected flows brian...

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