TURBULENCE MODELING IN PHOENICS

• The purpose of this 2 hour class is to give basic directions to

undergraduate students to use Phoenics to simulate turbulent

flows.

• The lecture gives a brief introduction to turbulent flows

features, to scales and to the law of the wall.

• The lecture applies to RANS models (algebraic and two

equation models) and the focus is on the selection of a proper

grid size near the wall to have a successful turbulent flow

simulation.

Turbulent flow features

• Fluctuatiation or Irregularity: turbulent

flow is random ie not deterministic.

The velocity fluctuates in all three

directions, so does the pressure and

temperature.

• Turbulent flow is intrinsically transient.

• Increased exchange of momentum: Diffusivity - spreading rate of jets,

boundary layers etc.

• Large Reynolds numbers;

• Dissipation of kinetic energy to internal energy

• Wide range of time and length scales

• Almost all practical flows are turbulent.

Turbulent flow features: Eddies Side view of a turbulent boundary layer Water jet at symmetry plane, Re 2300

• Eddies are "packets" of fluid (identifiable flow structures) with different

sizes ranging from macroscopic dimensions to the microscopic scale also

known as Kolmogorov scale.

– The large eddies are responsible to the transport of mass, momentum and heat.

– The smallest eddies dissipate kinetic energy into heat through viscosity

• The largest eddies scale to the characteristic flow dimension.

– Boundary layer ~ boundary layer thickness;

– pipe flow ~ pipe diamenter;

– jet flow~ jet width and so forth.

Kolomogorov (1941) theory: the energy cascade

Large eddies,

contain kinetic energy

Small eddies dissipates energy

into heat (viscosity)

“Big-size whirls have little whirls that fed on their velocity

Little whirls have lesser whirls and so on to viscosity”

• The energy is transferred from the mean flow to produce the large eddies.

• The energy of the large eddies feed smaller eddies and these in turn transfer

energy to smaller eddies yet.

• This process results in a transfer of energy in the form of a cascade from

larger eddies to smaller ones.

• The smaller eddies dissipates the energy into heat due to the viscosity action.

• The energy cascade suggest the multiple scales (size and frequencies) present

in turbulent flow. This feature difficults the development of turbulence models.

Kolomogorov (1941) theory: length scales • Turbulence phenomenon has multiples scales: the largest contain energy and

the smallest are responsible to dissipate energy.

• The scales refers to the time, frequency spectrum or length sizes.

• The ratio between the largest ‘l ‘ to the smallest scales ‘lk ‘is:

3 4

w

k

uRe where Re and u

Re

(U.l/)

Rel

(u*l/)

No

Nodes

2.00E+04 116 1.58E+06

1.00E+06 1623 4.28E+09

• Consider the turbulent flow in a plane channel:

• For Re of 2.104 it is necessary a million nodes grid to a channel volume

equivalent to a one hydraulic diameter; for Re of 106 are necessary a billion

nodes grid!

• The length scale ratio impacts on the grid size definition necessary to

encompass the largest to the smallest length scales.

The eddies’ sizes and Re

• The images below displays a jet flow with Re of 400 and 20000.

Observe as Re increase the refinement of the length scale of the

eddies due to Kolmogorov scale law: 3 4

k Re

• The eddies’ smallest scale lk ratio for Re of 400 and 20000 is of 19:1

l h = l/ReL3/4

Direct numerical simulation (DNS)

Large eddy simulation (LES)

Reynolds averaged Navier-Stokes equations (RANS)

Prediction Methods

The focus of this presentation rests on RANS models only

Numerical Simulations : estimates of cost

of fixed wing calculation

Sample fixed wing of AR=10, Re=5E+06

9

Boussinesq hypothesis

• Using the suffix notation where i, j, and k denote the x, y, and z

directions respectively, viscous stresses are given by:

• Many turbulence models are based upon the Boussinesq (1877)

hypothesis: Reynolds stresses are linked to the mean rate of

deformation.

• Exceptions are DNS and partially LES.

i

j

j

i

ijijx

u

x

ue

jiij i j t

j i

UUu 'u '

x x

10

Predicting the turbulent viscosity

• The turbulent viscosity can be estimated by algebraic models or by

solving a system of PDE.

• Algebraic models (mixing length and LVEL) employs expressions

similar to: T = C(dU/dy). They are computationally cheap but limited

to very simple flows (pipe and boundary layers)

• The 2nd category usually solves two equations to get T . So they are

called by two equation models: k-, k-, k-l, k- RNG, etc. For k-:

T = Ck2/ ; T derived by other 2 eq. models similarly follows k-.

• There is also the Reynolds stress model which instead of estimate T it

solves a system of six turbulent stress equations in order to get the

Reynolds turbulent stress tensor.

• The two equation models are largely applied to solve industrial

problems. Very often the constants are tuned to specific cases to

increase the accuracy.

11

Comparison of RANS turbulence models

Model Strengths Weaknesses

Spalart-

Allmaras

Economical (1-eq.); good track

record for mildly complex B.L.

type of flows.

Not very widely tested yet; lack of submodels

(e.g. combustion, buoyancy).

STD k-

Robust, economical, reasonably

accurate; long accumulated

performance data.

Mediocre results for complex flows with severe

pressure gradients, strong streamline curvature,

swirl and rotation. Predicts that round jets spread

15% faster than planar jets whereas in actuality

they spread 15% slower.

RNG k-

Good for moderately complex

behavior like jet impingement,

separating flows, swirling flows,

and secondary flows.

Subjected to limitations due to isotropic eddy

viscosity assumption. Same problem with round

jets as standard k-.

Realizable

k-

Offers largely the same benefits

as RNG but also resolves the

round-jet anomaly.

Subjected to limitations due to isotropic eddy

viscosity assumption.

Reynolds

Stress Model

Physically most complete model

(history, transport, and anisotropy

of turbulent stresses are all

accounted for).

Requires more cpu effort (2-3x); tightly coupled

momentum and turbulence equations.

Wall bounded flows

• The presence of a wall damps the velocity and the turbulence near the

wall region.

• To capture the wall effect the turbulence models have to be modified

near the wall.

• The embodied wall models on the turbulence models are grid

dependent.

• The success of turbulent flow simulation starts on a careful choice of

the grid spacing near the wall.

• This presentation focus on near wall grid spacing selection to a

successful simulation. To beginners this a potential area to make

mistakes

Wall bounded flows and velocity damping near the wall

Experimental turbulent boundary layer

vel. Profiles for various pressure

gradients. Coles 1968

• The presence of a wall damps the

velocity and the turbulence near the

wall region.

• To capture the wall effect the

turbulence models have to be

modified near the wall.

• The success of turbulent flow

simulation starts on a careful choice

of the grid spacing near the wall.

The meaning of the layers

• Inner layer: is the layer closest to the wall and is ruled by the fluid

viscosity. The viscous length scale /v* is greater than the wall

distance, or yv*/ = y+ << 1. On this region, u+ = y+ or = u/y. For

modeling purposes the inner layer extends up to y+ = 5

• Overlap layer: is the region where the viscous action start loosing

influence and the inertial effects increases. For modeling purposes this

layer ranges from y+ = 5 up to 40.

• Log layer: it is sufficiently far from the wall such that it is completely

rulled by the inertial effects but close enough to the wall that the shear

stress is uniform and equal to the wall shear stress! Usually the log

layer ranges from y+ = 40 up to 200

• Outter layer: flow dominated by the eddies inertial effects, y+ > 200.

Inner + Buffer + Overlap Layers

The velocity profiles near the wall are universally expressed in a

dimensionless form:

*

*

*

W

y vy

uu

v

where

v

is the eddy

velocity scale.

Outter Wall

layers Inner

5

Overlap

40

Log

200

*

uu

v

*y vy

Overlap: sensitivity to adverse pressure gradient

Phoenics turbulence models

• Algebraic models: constant turbulent viscosity, mixing length, lvel.

Lvel is the most popular, it can be applied starting from the inner layer

or from the log layer. It is an algebraic expression which fits the near

wall flow. These models are computationally cheap and delivers good

results for channel or boundary layer flows.

• Two equation models (standard) : refers to the k-e and variants. The

distance from the wall of the first volume center has to lay within the

log layer, 40 < y+ < 200.

• Two equation models (low Re) : low Reynolds flows usually have the

first node distance for 40 < y+ < 200 corresponding to a significant

percentage of the domain (> 15%). To avoid numerical errors induced

by the coarse grid it is preferred to starting integrating the flow

starting from the wall. The first node must lay at y+ ~ 1 and at least 3

more volumes up to y+ < 5. This assures a smooth integration along

the inner, overlap and log layer.

How to get good results from turbulence models

• All models have limitations, one can not ask more than the model

can deliver.

• Algebraic models are the simplest and the most limited. Usually

work well for pipe flows and boundary layers.

• 2 equation models are more accurate than algebraic models but

their constants do not have universally, it is possible for certain types

of flow a better matching by tuning the constants.

• 2 equation models considers flow in equilibrium: all the

production of k is dissipated at the same point. Flow with sudden

changes in direction are prone to be out of equilibrium.

• In general, turbulent models demand a special care with the near

wall mesh. If you get this wrong your simulation will fail.

WKSP #1-Turbulent flow develop. 2D channel

• This WKSH deals with a turbulent flow development along a 2D

channel flow.

• The flow simulations will be done using Parabolic model (visit (1)

and (2) links to further information).

• Parabolic model apply only to one-way flows (no recirculation

zones present). It will used along the WKSH because it is far more

efficient than Elliptic model.

• Along the WKSH will be supplied specific hints to set up a problem

employing Parabolic model.

2D Channel dimensions & properties

(a) For parabolic model the main flow direction has to be aligned with the Z axis.

(b) Parabolic flow has w component always along z > 0 direction, if happens to have

w along z<0 it will fail. Due this feature it does not need specification of an outlet.

(c) Parabolic model visits only one slab at a time and stores only the last slab. To

provide full field storage go to VR-EDITOR box GRND and set IDSPA=1, IDSPB=1

and IDSPC=80 (always equal to NZ), IDSPD=1

L = 10m

Inlet

z (a)

y

H/2=0.05m

(b) no outlet

specification

Uniform GRID (c)

NX=1

NY=28

NZ=80

nwall (plate)

Center line

Properties

(~air)

= 1.0 kg/m3

= 10-5 m2/s

Objects

Inlet: Win = 10 m/s

Re = 2.105

Inlet: 5% turbulence

Plate: no slip, W=0

Numerics

Lsweep = 100

Relax (manual)

Output

Pause at end of run

Storg.: YPLUS & STRS

A C C E S S T H E V R - E D I T O R B O X E S A N D S E T:

Models

K-E

WSTRS

WYPLUS

Phoenics variables

2H URe

2D Channel Results –

W velocity at z/D of 2D, 10D and 40D

One may reproduce this figure using autoplot or get similar results

employing ‘Ploting variable’ in VRVIEWER.

2D Channel Results – W center-line velocity

overshoot

experimental

overshoot,

see link

16D 32D 48D 64D 80D

2D

Ch

an

nel

Res

ult

s –

Y

+ &

ST

RS

1st volume distance correct: 40 < Y+ < 100.

1st volume within Log Layer!

16D 32D 48D 64D

STRS = w/

STRS fully developed = 0.192

STRS Colebrook-White = 0.195 (1.5% off)

2D

Ch

an

nel

Res

ult

s –

T, k

an

d

Viscosity is a flow property: T = Ck2/

Near the wall the ‘turbulent viscosity’ is 16 times

greater than the molecular viscosity.

The largest changes on k and are near the wall.

At the centerline T is nearly 300 greater than .

16D 32D 48D 64D 80D

experimental

overshoot, ENUT

KE

EP

Phoenics adjustable constants for KE model

• The choice of one constant value by other depends on previous

knowledge about a specific flow and also knowledge about the

meaning of the constants within the model, for KE visit turbulence.

Exp

lori

ng t

he

gri

d s

ensi

tiv

ity

to

Y+

an

d S

TR

S • Evaluating Cf thru Colebrook-White is possible to estimate, at the fully

developed region: (i) 1st node distance from the wall (), (ii) number of volumes

NY (uniformly distributed), (iii) STRS = w/. The last will be used to check the

numerical solution. Keep in mind that H/2 = 50mm.

KE – Low Re

KE – Low Re

KE - Standard

KE - Standard

2

W fU CSTRS

2

Workshop#1 – adjusting grids

• Based on the table below select between ‘standard’ or ‘low Re’ KE

models to cases where W = 1 and 50 m/s, also define a new NY. Do

these changes on the previous Q1.

• For reference, the previous case velocity is W = 10m/s or Re 2E+05,

• STRS Cole is the w/ value estimated by Colebrook-White for a fully

developed flow.

• Y+ = 1 thru 100 are the estimated first node distance to the wall.

W Re2H Cf Cole STRS Cole Y+=1 Y

+=40 Y

+=100

(m/s) (---) (---) (m/s)2 mm mm mm

1 2.00E+04 0.0065 0.0032 0.18 7.03 17.58

10 2.00E+05 0.0039 0.1954 0.02 0.90 2.26

50 1.00E+06 0.0029 3.6389 0.005 0.21 0.52

Workshop – adjusting grids

• Case W = 1m/s – ‘Low Re’ because the 1st volume for y+ = 40 would

be at 7mm from the wall which is nearly 14% of the Y length. One can

use a uniform grid or a more economical 2 region grid, the near wall

grid extends up to y+ = 40 (~7mm)

• Case W = 50m/s – Standard,

W Re2H KE NY Q1

(m/s) (---)

1 2.00E+04 LowRe 142 unif 2 reg

10 2.00E+05 Standard 28 unif

50 1.00E+06 Standard 48 unif ellip

Workshop#2 – Flow in backward face step

• A simple backward step flow challenges the 2 equation turbulence

models because it has a shear layer, a recirculation zone, null wall

shear stress point and a redeveloping boundary layer!

• Compare the reattachment point position employing: k-e model, the

Chen-Kim k-e model, the RNG k-e model, the k-omega model and the

LVEL model.

Workshop#2 – Flow in backward face step

• Load case T103 from library. Step height, H = 0.038m and ReH =

45000.

H

Y

3H

X

L = 0.762 m or 20H

s = 0.1524 m or 4H

Uin = 13 m/s

• Change X grid regions to 10 and 60 cells (no power)

• Change Y grid regions to 8 and 10 cells (the last with pwr 1.5 sym)

• Start with the KE standard model

Workshop#2 – Flow in backward face step

• Key flow features to

explore are the wall shear

stress and if possible the

pressure distribution at the

channel bottom wall.

• The reattachment point

is determined by the x

distance where the wall

shear is null. stress is null.

Workshop#2 – Flow in backward face step

• Fill in the table

x (m) x/H

k-e

Chen-Kim k-e

RNG k-e

k-omega

LVEL

Workshop#2 – Flow in backward face step

• As the reattachment point is approached w 0 and Y+ 0

• Would you consider Low Re type models to capture the

reattachment point?

• If so use k-e low re, same grid as before but modify Y 1st region to

60 cells, use pwr of + 1.6 and in Numerics set to 10000 sweeps

• Discuss the validity

of the results in view

(k-e model) of the y+

values

Lower bound to two eq. Standard models

Workshop#2 – Flow in

backward face step

• The k-e low re apparently

estimates nearly the same

reattachment point position as

predicted by k-e standard!

• But the magnitude of the

STRS has changed

significantly!

• In fact the k-e standard made

wrong STRS estimations

because it employed an

inappropriate grid which

resulted in y+ values lower than

40 at the neighborhood of

reattachment point!

END