One-Dimensional Stochastic Simulation
of Turbulence-Microscale Interactions
Alan Kerstein
Combustion Research Facility
Sandia National Laboratories
Livermore, CA
Contributors:
Wm. T. Ashurst, T. D. Dreeben, T. Echekki, J. C. Hewson,
V. Nilsen, R. C. Schmidt, S. E. Wunsch
January 28, 2002
IPAM Workshop
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Outline of presentation

One-dimensional models of turbulence, past and present

Overview of the present approach

Representative applications
– Planar mixing layers
– Buoyant stratified flows
– Combustion

Near-wall momentum closure for large-eddy simulation (LES)

A bottom-up approach to multidimensional turbulent flow simulation

Conclusions
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Use of the boundary layer approximation and the
mixing length hypothesis in turbulence models
Example: temporal planar boundary layer
Laminar:
y
Re  UL /
ut   u yy
L(t )
Turbulent ( Re  1):
ut   e u yy
Mixing length hypothesis:
 e ( y) ~ y 2 /
where
u (0, t )  U
1/  ~ u y
Quasisteady assumption:

Lateral momentum flux  e u y is independent of
2 2

Then y u y  constant, giving u ~ log y
U
y in some y range
Limitations:

Describes mean flow

Parameterization of  e is flow dependent

Advection is modeled as diffusion, affecting the realism of couplings to microphysics
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The mixing length concept is not inherently
restricted to average behavior
‘Typical’ eddy at height y:
u( y2 , t )
y
y
u ( y1, t )
This picture motivates a time-resolved 1D model
involving a sequence of fluid displacements
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A time resolved formulation can have a
range of eddy sizes crossing a given height
Why include all eddies?

Averaged models sometimes
allow for more than one eddy
size, e.g., ‘inactive’ (large)
eddies in boundary layers

Desire an eddy population
governed by the ICs and BCs
(rather than flow specific
parametric assumptions)
y

Resolve small eddies to obtain
realistic advective-diffusive
coupling, e.g.,
– viscous wall layer
– diffusive interfaces in buoyant flows
– thin flames in turbulent combustion
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Goals

Predict or explain properties of flows such as
– Wall boundary layers
– Planar shear flows (mixing layers, jets)
– Horizontally homogeneous buoyant stratified flows
(e.g., many geophysical flows)
– Cylindrically and spherically homogeneous flows
(pipe flow, stellar convection)

Develop improved subgrid-scale (SGS) models
for 3D simulations (e.g., LES)
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A new modeling approach:
One-Dimensional Turbulence
Molecular evolution based on a boundary layer
formulation is supplemented by an ‘eddy process,’
e.g.,
ut   u yy  eddies
To specify the eddy process, need


Definition of an eddy (biography)
Eddy selection procedure (demography)
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1D eddy implementation involves two steps
Kinematics:
Lateral fluid displacements with properties
(velocity, etc.) unchanged within fluid elements
Dynamics:
Modification of velocity profiles within the
eddy to reflect


conversion between kinetic and
gravitational potential energy
energy exchange among velocity
components (pressure scrambling)
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The triplet map is a 1D procedure
TRIPLET MAP: A "SHELL GAME"
that emulates
3D eddy kinematics
TRIPLET MAP
The triplet map captures
compressive strain and
rotational folding effects,
and causes no property
discontinuities
c(x)
The triplet map
is implemented
numerically as
a permutation
of fluid cells
c(x)
x
This procedure
emulates the
effect of a 3D
eddy on property
profiles along a
line of sight
c(x)
x
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The triplet map transports momentum
laterally and amplifies shear
High shear at small scales drives small eddies, leading to an eddy cascade
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Eddy dynamics
Wavelets are added to velocity profiles within
an eddy to change kinetic energy components
while satisfying conservation laws
Wavelets measure or modify fluctuations on
a particular scale (e.g., the eddy size). Unlike
Fourier transforms, wavelets are applied
locally (e.g., to an eddy) rather than globally.
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An eddy event event consists of a triplet
map followed by velocity profile changes
20
20
u
v
w
c
15
15
10
Triplet map
is applied to
all properties
(velocities
u, v, w and
scalar, c)
5
0
-5
5
0
-5
-10
5
10
y
0
20
15
5
10
15
20
20
15
Dynamic
step affects
velocities but
not scalar
10
u, v, w, c
-10
0
eddy range
u, v, w, c
u, v, w, c
10
5
0
-5
-10
0
5
10
15
20
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Eddy selection procedure


Assign a time scale  to each possible eddy,
parameterized by eddy size and location within the
computational domain
The set of  values defines an eddy rate distribution l
from which eddies are sampled
The physics is in the determination of 
(as in mixing length theory)
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Determination of  (schematic)
Principle:
Enforce consistency of eddies and flow (velocity and density profiles)
Eddy:
Eddy velocity ~ l /  so eddy energy ~  l 3 /  2 ( l  eddy size)
Flow:
P  gravitational potential energy change caused by eddy
K  maximum kinetic energy extractable by adding wavelets
to velocity components
Relation determining  :
 l3

2 

 A  K  P  Z
l

2
adjustable parameters
viscous penalty (imposes a
threshold Reynolds number)
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This approach is much like conventional
mixing length theory
but

The concept is applied to all l values, not to a single l value

K and P are computed using the instantaneous state, not
an average state

Concurrent eddy and molecular processes are strongly
coupled, thereby linking eddy dynamics to the flow
configuration (ICs, BCs, body forces, fluid properties, etc.)
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Simulated eddy time histories reflect large
and small scale flow evolution
Temporal planar free shear flows:
mixing layer
wake
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One-Dimensional Turbulence (ODT)
captures fundamental turbulence properties
Homogeneous turbulence:

Power spectra
– Inertial and dissipative subrange scalings (Kolmogorov)
– Pr-dependent passive-scalar subrange scalings (Batchelor, etc.)
– Buoyancy-driven scalar spectrum (Bolgiano-Obukhov)

Decay of grid turbulence

Scalar mixing (variance decay, PDF evolution)
Inhomogeneous turbulence:
The focus of this presentation
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ODT simulation of variable density
spatially developing mixing layers

Step function initial profiles of density and
streamwise velocity

Flow parameters
– Free stream velocity ratio: r = u2/u1
– Free stream density ratio: s = 2/1
(u1 > u2)

Boundary layer equations for spatially developing
variable density flow, with suitable lateral boundary
conditions

Generalized eddy process incorporating
– Density variations
– Conservation laws for spatially developing flow
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Mean profiles for r = 0.38: comparison of ODT
and measurements (one adjusted parameter)
s = 1/7
s=1
s=7
curves: ODT
symbols: measurements by Brown and Roshko (1974)
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ODT reproduces s dependence of growth
rate and suggests a large-s transition
Scaled growth rate = width / downstream distance
width 
2
 u1  u~ )(u~  u2 ) dy

( 1   2 )(U ) 2
width 
1
u1  u~ )(u~  u2 ) dy

(U ) 2
u
u~ 

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Interpretation:
high density contrast inhibits layer growth
y
1
unrestricted entrainment
M
2
restricted entrainment

• Shear resides in the mixed layer
• Energy of entraining eddies scales as M
• For 2 >> M, the energy requirement for entrainment of fluid 2
restricts heavy-side intrusion
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Implied density profile for 2 >> 1
1
y
smooth transition
rms
sharp interface
peak due to flapping
of sharp density gradient

2
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ODT and measured (Konrad 1976) scalar
fluctuation profiles support this picture
passive scalar,
s=1
normalized density fluctuations,
s=7
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ODT prediction: increasingly sharp density
interface as the density ratio increases
mean profiles (dashed),
instantaneous density profile (solid)
normalized density fluctuations
r = 0.38
r = 0.38, s = 7
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A zonal picture suggests an interpretation of
asymmetric s dependence of the growth rate
light
mixed
heavy
u1
uM
dM
u2
1. Hypothesis: Spatial growth is fastest if uM is slow
2. Mixed zone velocity is dominated by heavy fluid
(momentum is density-weighted velocity)
3. By 2., uM is slow if heavy fluid is slow, giving fastest growth (by 1.)
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Implications

High density contrast introduces an inertial barrier
to entrainment

Is this a significant mechanism of entrainment
suppression across stably stratified interfaces?
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Rayleigh-Benard convection is a simple flow
configuration with complicated behaviors
Imposed
temperature
difference
T1 > T2
T2
mean
temperature
mean
density
T1
Mean heat flux is independent
of height, so mean temperature
gradient is high where thermal
diffusivity is low (near walls).
Far from walls, eddy diffusivity
dominates.

Parameters:
– Ra: buoyant forcing scaled by viscous damping
– Pr: viscosity scaled by thermal diffusivity

Measured properties:
– Nu: heat flux scaled by heat flux in motionless fluid
– Temperature and velocity fluctuations (variance, PDF)

Classical analysis (Priestley):
– Near-wall flow unaffected by opposite wall
– Implication: Nu ~ Ra1/3

Observation: Scaling exponent depends on Ra and Pr
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Nonclassical scaling implies
wall interactions
Two possible mechanisms:

Coupling via the mean density gradient in the
central region

Large scale motions
Large buoyant plumes
and large scale circulation
are seen in experiments
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Simulated Rayleigh-Benard density profile
for Ra = 1.4 * 10 9, Pr = 0.7
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Measured heat transfer (curves) is reproduced
(symbols) by adjusting two ODT parameters
Pr = 0.7
Nu Ra1/ 3
Pr = 2750
Pr = 0.025
Ra
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Computed midplane density fluctuations match
Pr = 0.7 data (curve) and predict Pr trend
Pr = 0.025
Pr = 0.7
rms / 
Pr = 2750
Ra
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Scaled ODT centerline density PDFs for
Ra = 1.4 * 10 8; expt. (Chicago) for Ra = 9 * 1011
rms p(  )
Scaled PDFs (ODT and
measured) are insensitive
to Ra. ODT Pr dependence
has not yet been tested
experimentally.
(    ) / rms
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Interpretation of ODT results: Two mechanisms
contribute to core density fluctuations
Mechanism
Regime of
Dominance
Implied scalings
Nu
rms
Stirring of mean
density gradient
by core eddies
low Pr
nonclassical
classical
Entrainment of
unmixed wall-layer
fluid into core
high Pr
classical
nonclassical
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ODT is being used to study various
environmental stratified-flow phenomena

Penetrative convection (Deardorff-Willis experiment)

Shear driven and convectively driven entrainment at stable density interfaces

Stratified boundary layers

Layering in stably stratified flow (Phillips mechanism)

Mixing-induced buoyancy reversal (cloud-top entrainment instability: Randall)

Plume dispersal

Mixing-controlled droplet growth in clouds

Nuclear burning coupled to convection (supernovae)

Multicomponent convection:
– semiconvection (stars)
–
thermohaline staircase (oceans)
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A slow-diffusing stable species can cause layering
of a convection process: double-diffusive instability
S
cold, fresh

T
solid: initial state
dash: later state
(salt diffusivity
assumed negligible)
warm, salty
S
T
S
T
S
 S T
T

convective layer
forms, then heat
diffuses across
stable interface,
initiating a new
layer:
thermohaline
staircase
heat flux
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Comparison of ODT and measured staircase
development (no parameter adjustment)
Huppert and Linden (1979)
ODT
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The ODT final state reflects the wide range of
dynamically relevant time and length scales
temperature
salinity
density
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Flux ratio (salinity/temperature)
across interface
ODT captures the observed regimes
of diffusive interface structure
curves:
symbols:
measurements
computations for two
flow configurations
Ratio of density jumps (salinity/temperature)
across interface
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Spatial structure of a simulated jet diffusion
flame resembles flame images
Simulation of a piloted methane-air flame (Sandia flame D)
One realization
(horizontal
segments
denote
eddy events)
average of 100
realizations
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ODT resolves advective-diffusive-reactive
couplings and hence all flame regimes
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A near-wall momentum closure for LES: full ODT
in wall layer, ‘filtered’ eddies in nearby layers
Y
NODT
LES/ODT overlap region
L
inner
region
an eddy event
here is not
allowed
Y
Y
all eddy
events must
extend into the
inner region
no-slip
wall
Z
X
Illustration of the allowable locations
of ODT eddy events
ODT sub-control volumes imbedded in an
‘inner region’ LES control volume

ODT sub-control volumes are coupled vertically and laterally, reflecting continuity and momentum constraints

ODT advective transfers (eddy events) across LES cell faces determine vertical fluxes for LES closure

ODT responds to global flow evolution through coupling to the LES pressure update
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LES channel simulation using ODT near-wall
momentum closure for Re = 1200
25
LES
LES
LES
LES
ODT
recent small
eddy events
streamwise velocity, u
20
15
time average profile
instantaneous profile A
instantaneous profile B
10
sharp
smooth
5
0
0.7
0.75
0.8
0.85
0.9
y/h (distance from centerline)
0.95
1
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Comparison of DNS (Moser et al.) and
LES/ODT mean velocity profiles
25
Re = 590 LES/ODT
Re = 1200 LES/ODT
Re = 2400 LES/ODT
Re = 590 DNS
20
u+
15
10
5
0
1
10
100
y+
1000
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Friction law
Solid line: experimental correlation (Dean)
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Velocity fluctuations for Re = 590
Filled symbols:
Open symbols:
Curves:
ODT (v and w statistics are identical)
LES/ODT
DNS (Moser et al.)
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ODT (without LES) reproduces key features
of mixed-convection boundary layers

Horizontal surface: ODT reproduces
– Monin-Obukhov (MO) similarity scaling (transition from
near-wall shear dominance to far-field buoyancy dominance)
– MO similarity functions measured in the atmos. bdy. layer

Vertical surface: ODT reproduces and extends nonclassical
scalings seen in DNS (Versteegh and Nieuwstadt)
u
heated
wall
T T
y
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ODT motivates a new
3D turbulent flow simulation concept

Form a 3D lattice of linear domains

Flow is driven by transfers of fluid between domains
with different orientations: each transfer forces flow
contraction (dilatation) on the donor (receiver) domain

Rules for transfer are coupled to viscous evolution
through dependence on the instantaneous flow field

If the lattice resolves the flow, this (ideally) becomes a
Navier-Stokes simulation driven by local interaction
rules (as in lattice gas hydrodynamics)

On an under-resolved lattice, the ODT eddy process is
the SGS advection model
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Possible applications of ODT
to geophysical flows
Troposphere:
• ABL diurnal cycle, including characterization of variability
• ABL parameterization, including cloud effects
• Cloud microphysics (initial application by S. Krueger, Univ. of Utah)
• Cloud dynamics (e.g., cloud-top entrainment instability)
• Surface-layer dry deposition, including coupling to chemistry
• Local-source dispersion, including concentration fluctuations
Stratosphere:
• Stratified Kelvin-Helmholtz instability with ozone chemistry
Oceans:
• Species exchange at ocean-atmosphere interface
• Wind-driven surface-layer species and momentum transport
• Deep-ocean transport, including barriers to transport
General:
• ODT-based Single Column Model for use as a near-surface SGS model in a
General Circulation Model
• Simulation tool for planning and interpretation of microstructure measurements
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Conclusions

The phenomenology of the fundamental turbulent
flows is neither fully known nor fully explained

ODT is both a tool for studying turbulence
fundamentals and a bridge between fundamentals
and more complicated cases
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