Andrew T. Norris

Ohio Aerospace Institute, Brook Park, Ohio

Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Turbulent Chemical Interaction Models in NCC:

Comparison

NASA/TM—2006-214417

December 2006
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Turbulent Chemical Interaction Models in NCC:

Comparison

NASA/TM—2006-214417

December 2006

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Andrew T. Norris

Ohio Aerospace Institute, Brook Park, Ohio

Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio
Acknowledgments

The authors wish to thank T.-H. Shih who meshed the grid for the initial one-element case, and also performed a series of

calculations using that geometry.

Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22161

Available electronically at http://gltrs.grc.nasa.gov

Level of Review: This material has been technically reviewed by technical management.

This report contains preliminary findings,

subject to revision as analysis proceeds.
Abstract

The performance of a scalar PDF hydrogen-air combustion model in predicting

a complex reacting flow is evaluated. In addition the results are compared to

those obtained by running the same case with the so-called laminar chemistry

model and also a new model based on the concept of mapping partially stirred

reactor data onto perfectly stirred reactor data. The results show that the scalar

PDF model produces significantly different results from the other two models,

and at a significantly higher computational cost.

Introduction

The simulation of turbulent reacting flow

by RANS and LES requires the use of many

models to solve for various processes. One

of the most important, but often most

neglected is the model for the interaction

between turbulence and the chemistry of

combustion. Previous work [1] has shown

that the errors in the chemistry source

term caused by not using a model can be

of several orders of magnitude. However

models for this phenomenon are often

costly in CPU time and make obtaining a

solution very costly. However with

advances in computational power, the use

of even the most complex of models is

now possible for realistic geometries.

In this study, a nine-element hydrogen

burner is simulated, using laminar

chemistry, a new reactor mapping model

and a Scalar PDF model. By comparing the

results it is hoped to demonstrate both the

capability to perform high-fidelity

simulations using these models in NCC,

and also compare the results with each

other.

Laminar Chemistry

The simplest of turbulent-chemistry

interaction models and also the most

commonly used is the “Use No Model”

technique. In this we assume there is no

interaction between the chemistry and

turbulence, and as a consequence, simple

laminar kinetics are used. This simulation

will be the baseline for comparing both the

timing and results of the other models.

PDF Model

Almost all codes in use by industry today,

are based on solving the mean NavierStokes and Species transport equations,

with models for the unknown terms such

as the k-e model. These codes represent

the flow quantities at discrete locations by

a single number: the mean. However if one

was to take an experiment, and run it

many times, obtaining the flow value at

some set time, each value would be

different. Collecting enough samples this

way would give you the distribution of the

property, from which one could obtain the

PDF (Probability Density Function) of the

Turbulent Chemical Interaction Models in NCC: Comparison 

Andrew T. Norris 

Ohio Aerospace Institute 

Brook Park, Ohio 44142 

Nan-Suey Liu 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

NASA/TM—2006-214417 1
property at that location. Clearly the PDF

provides far more information about the

flow than just the mean value. From the

PDF, all other moments can be obtained,

such as variance, skewness, flatness etc.

And if more than one property is recorded,

you have a joint PDF of the properties,

from which correlations can be calculated.

PDF methods then, are based on solving

the transport equation for the joint PDF of

the properties of interest. [2,3] By doing

this, a more detailed description of the

flow can be obtained. For example, if we

were to solve the transport equation for

the joint PDF of the three velocity

components, U, V and W, we would have

the exact value of such correlations as

<uv>, <uuu> and <uw> available, and so

would not need to calculate separate

transport equations for Reynolds stresses.

(The angled brackets denoting mean

quantities, lower case symbols the

fluctuating quantities).

However the real advantage of PDF

methods is in turbulent reacting flows. In

the source term for the species transport

equations, we have the reaction rate of the

i'th species, S_{i} as a function of all the

species ø_{1..n}, and state properties T, P.

S_{i} = S_{i}(ø_{1..n}, ,T, P) - (1)

This term has a big effect on the

calculations, as it is responsible for large

changes in the density and viscosity of the

fluid, and so has a large effect on the

velocities.

When one solves the equations for a mean

Navier-Stokes code, this source term

becomes the mean reaction rate: <S_{i}>

<S_{i}> = <S_{i}(ø_{1..n},T, P)>

 ≠ S_{i}(<ø_{1..n}>,<T>,<P>).– (2)

The inequality in Eqn. 2. is significant,

often of several orders of magnitude, and

so the reaction rate term has to be

modeled. However, the complexity of the

term, as well as the need for a different

model for each different species and

property combination makes modeling

unattractive.

In the PDF approach however, this term

does not need to be modeled. The PDF of

the species and state variables are known,

and so this term is treated exactly.

Comparisons between PDF predictions and

those calculating the mean reaction rate

with Eqn. 2 show errors in temperature of

500K and significant errors in the flow

field.

However there are terms that do require

modeling in the PDF transport equation.

Depending on the properties being solve

for, these include the molecular mixing,

turbulent diffusion and fluctuating

pressure term. Of these, the molecular

mixing term is the area of most

uncertainty. However the effect of these

models on the flow field is secondary to

the effect of the reaction rate, which

makes the PDF method very attractive in

solving turbulent reacting flows.

Reactor Mapping Model

For a finite volume, RANS CFD code, the

modeling of the combustion process in a

single cell can be thought of as that of a

perfectly stirred reactor. There is a mass

inflow of certain mean species

concentrations, which are mixed

instantaneously with the mean gas mixture

contained in the cell. The resulting

NASA/TM—2006-214417 2
averaged mixture is then reacted and the

resulting mean products are what exit the

cell.

Considering the more sophisticated scalar

PDF model, one can again liken the

process of a single cell to that of a simple

reactor: In this case a Partially Stirred

Reactor (PaSR). For the PaSR, a distribution

of species enters the reactor or cell. There

they are mixed via the turbulent process

while reacting. And the resulting

distribution of products is what exits the

reactor, or cell. And in each case, the

mean species concentrations and

temperatures reflect the physics of the

process better than the simple Perfectly

Stirred Reactor (PSR) results.

To this end the reactor mapping model

(RMM) was developed as a mapping

between the PaSR results and the PSR

results. PaSR temperature results for a

given residence time and turbulence level

were mapped onto the temperature results

for a PSR using the residence time, R_t, as

the mapping variable. Then taking R_t and

multiplying by the turbulent frequency,

e/k gives a non-dimensional number R for

this mapping, and a simple functional

relationship between R and the equivalent

residence time, R_equiv, provides the

mapping function.

The function for the equivalent residence

time used in this study was:

R_equiv/R_t = 0.434 ln (R) - 0.176 - (3)

Similar functions exist for other fuels, and

so this model can be applied for most

cases.

To apply this model, the time used to

compute the change in scalar quantities

for a given computational step is adjusted

by Equation. 3. That is, while all other

processes in a given cell are incremented

by a time step R_t, the chemical source

term is only integrated over R_equiv.

Further details of this model are provided

in a separate paper [4].

Simulation

To demonstrate the effect of using a

turbulent combustion model in a realistic

simulation, the case of the Sandia 9

element hydrogen injector was chosen.

The simulation of the hydrogen burner

involved two separate computations. First

a simulation of just one nozzle was

performed, which included an upstream

settling chamber, and a downstream

expansion. The size of the settling

chamber and expansion chamber was

made to provide an identical expansion of

area as that of the nine-nozzle case. The

two hydrogen jets were provided with

plenums, to damp out any possible

fluctuations of the inlet conditions.

Construction of the grid used 493,015

elements and is shown in Figure. 1.

After the simulation of the single nozzle

case had run to convergence, data was

taken at a location 5mm. upstream of the

expansion region. This location is just

downstream of the hydrogen jets, (2.6

mm) and is used as the initial conditions

for the nine-nozzle simulation.

The quantities recorded at this location are

the profiles of the velocity, turbulence and

scalar variables. These fields are then

interpolated onto the inlet regions of the

nine-element simulation. Each of the nine

nozzles had the initial data orientated so

the hydrogen jets corresponded to the

NASA/TM—2006-214417 3
experimental location. The grid used

contained 282,624 elements and is shown

in Fig. 2.

For the scalar PDF simulation, 100

particles were used at each cell to

represent the joint scalar PDF. All

computations were performed on the

NASA Glenn Cortland cluster, using 32 or

64 processors.

For all cases a 9 species, 20 step chemistry

model for hydrogen-air combustion was

used, as described in Table 1.

Results

Timing results for the simulations are

shown below:

Simulation CPU Iter. Time

(seconds)

Iteration

time for

64 CPUs

Finite Rate 32 5000 10,428 1.04

RMM 32 5000 10,732 1.07

PDF Model 64 5000 153,830 30.76

As can be seen from the table, the

difference between the RMM and the

baseline Finite Rate model is negligable.

However the use of the Scalar PDF model

gives an increase of thirtyfold in the

required computational resources.

Fortunately the code does scale very well

for parallel work and so could be run on

about 1000 processors to achieve a similar

turn around time to the other two

simulations.

Looking at the temperature fields shown in

Figure 3, it is apparent that there is a

significant difference between the scalar

PDF results and the other two simulations.

The most obvious feature is that the PDF

results show a more persistent cool core

for all 9 injectors, while at the same time

being surrounded by a hotter reacting

layer. The laminar chemistry and PSR

results show a central core that mixes with

the surroundings quicker, and with a hot

reacting layer present only about the

central core. These temperature fields are

also reflected in the major products of

combustion data (not shown), which show

the same behavior.

Looking at a representative of the minor

species fields, OH, shown in Figure 4

shows a similar trend in the data.

The levels of OH reflect the structure of

the temperature fields shown in Figure 3.

The isolated hot cores of the laminar and

RMM results show high levels of OH, while

the PDF results show a more distributed

amount of OH at lower concentrations. In

this data the effect of the RMM can be

seen, in that it exhibits a trend toward the

PDF results, with a slightly more

distributed concentration of OH

throughout the burner.

Looking at the velocity fields in Figure 5

also shows the effect of the more

complete combustion in the PDF

simulation.

The velocity profiles show a significant

amount of variability, with the scalar PDF

results exhibiting significantly higher

velocities and also smaller regions of

recirculation. The higher velocities can be

attributed to the more complete

combustion and resulting expansion of the

gasses. This in turn tends to damp out

recirculation zones. The laminar and RMM

data show very little difference, with the

NASA/TM—2006-214417 4
RMM result having a slightly higher

maximum axial velocity.

It should be noted that no experimental

data is available so far for this

configuration. Hence no quantitative

comparision can be made as to which

model works best. However the assumed

PDF does provide significantly more

physics to the combustion model, and the

differences between the cases shown here

are intriguing.

What can be gained from this study is

firstly that the most physically complete

model for turbulent chemistry interaction

can be used to model realistic problems,

though at a significantly greater

computational cost than other models. In

addition the performance of the RMM is

interesting. Overall it showed a similar

performance to the laminar chemistry

model, however in the ways it did differ, it

was in a manner that trended toward the

scalar PDF results. As the RMM is very

much under development at the moment,

there is good reason to suppose that

further work might produce a model that

provides much of the physics of the scalar

PDF model, but without the high

computational cost.

Summary

The effect of the interaction between the

combustion chemistry and the turbulent

flow field is an important phenomena in

combustion CFD. Of the models available,

the scalar PDF model can be considered

the best, as it requires no modeling of the

chemistry source term. And with

computational resources now available, it

is possible to run realistic simulations

using the scalar PDF model. In an example

run of a 250,000 element simulation, the

scalar PDF model showed a very different

solution than that of a case with no

turbulent-chemistry interaction model or

the PSR model. Further development and

validation of the Scalar PDF model would

seem to be an important goal to enhance

the National Combustion Code and NASA

computational abilities. In addition the

promise shown by the RMM suggests that

this might, with further development,

provide a simple robust model for

turbulent chemistry interactions without

the high computational cost associated

with the scalar PDF model.

Bibliography

[1] A. T. Norris and A. T. Hsu. Comparison of

PDF and moment closure methods in the

modeling of turbulent reacting flows. In 30th

AIAA/ASME/SAE/ASEE Joint Propulsion

Conference, Indianapolis, Indiana,

AIAA–94–3356, 1994.

[2] S. B. Pope. The probablility approach to the

modeling of turbulent reacting flows.

Combustion and Flame, 27, 299-312, 1976.

[3] S. B. Pope. PDF methods for turbulent

reactive flows. Prog. Energy Combust. Sci., 11,

119-192, 1985.

[4] A.T. Norris, Reactor mapping model for

turbulent chemical interaction. Submitted AIAA

Reno Conference, January 2007.

NASA/TM—2006-214417 5
Figure 1. Single-injector geometry, showing upstream chamber to the left,

with two fuel ports with plenums and expansion chamber. Contours show

temperature for finite-rate simulation with no chemistry-turbulence interaction

model.

NASA/TM—2006-214417 6
Figure 2. Geometry of Sandia 9-element hydrogen burner. Length is

approximately 50mm and diameter is approximately 38mm. Each injector is

7.6mm in diameter, and 5.0mm long. This corresponds to having the inlet

located just downstream of the hydrogen injection ports shown in Figure 1.

NASA/TM—2006-214417 7
Table 1: Arrhenius constants for 9 species, 20 step hydrogen-air mechanism used in

study.

Reaction A n E

H+O2+M=HO2+M

H2O/18.6/ H2/2.86/

 3.61E17 -0.72 0.

H+H+M=H2+M 1.00E18 -1.0 0.

H+H+H2=H2+H2 9.20E16 -0.6 0.

H+H+H2O=H2+H2O

6.00E19

-1.25 0.

H+OH+M=H2O+M

H2O/5/

 1.60E22 -2.0 0.

H+O+M=OH+M

H2O/5/ 6.20E16

-0.6 0.

O+O+M=O2+M 1.89E13 0.0 -1788.

H2O2+M=OH+OH+M

1.30E17

0.0 45500.

H2+O2=2OH 1.70E13 0.0 47780.

OH+H2=H2O+H 1.17E9 1.3 3626.

O+OH=O2+H 3.61E14 -0.5 0.

O+H2=OH+H 5.06E4 2.67 6290.

OH+HO2=H2O+O2

7.50E12

 0.0 0.0

H+HO2=2OH 1.40E14 0.0 1073.

O+HO2=O2+OH 1.40E13 0.0 1073.

2OH=O+H2O 6.00E+8 1.3 0.

H+HO2=H2+O2 1.25E13 0.0 0.

HO2+HO2=H2O2+O2

2.00E12

 0.0 0.

H2O2+H=HO2+H2 1.60E12 0.0 3800.

H2O2+OH=H2O+HO2

1.00E13

 0.0 1800.

NASA/TM—2006-214417 8
Figure 3. Temperature fields in Kelvins from the top for the laminar chemistry,

PSR model and PDF model respectively. Data consists of a slice along the

centerline and a cross section at x = 0.04m.

NASA/TM—2006-214417 9
Figure 4. OH mixture fraction fields (from the top) for the laminar chemistry,

PSR model and PDF model respectively. Data consists of a slice along the

centerline and a cross section at x = 0.04m.

NASA/TM—2006-214417 10
Figure 5. Axial velocity contours, (from the top) for the laminar chemistry, PSR

model and PDF model respectively. Data consists of a slice along the centerline

and a cross section at x = 0.04m. Units are meter/second.

NASA/TM—2006-214417 11
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E–15697

WBS 984754.02.07.03.06.01

17

Turbulent Chemical Interaction Models in NCC: Comparison

Andrew T. Norris and Nan-Suey Liu

Scalar PDF; Hydrogen combustion; Turbulence-chemistry interaction; National

combustion code

Unclassified -Unlimited

Subject Categories: 34 and 64

The performance of a scalar PDF hydrogen-air combustion model in predicting a complex reacting flow is evaluated. In

addition the results are compared to those obtained by running the same case with the so-called laminar chemistry model

and also a new model based on the concept of mapping partially stirred reactor data onto perfectly stirred reactor data. The

results show that the scalar PDF model produces significantly different results from the other two models, and at a

significantly higher computational cost.

Andrew T. Norris, Ohio Aerospace Institute, 22800 Cedar Point Road, Brook Park, Ohio 44142; Nan-Suey Liu, NASA

Glenn Research Center. Responsible person, Andrew T. Norris, organization code RTB, 440–962–3071.