One-Dimensional Stochastic Simulation of Turbulence-Microscale Interactions

1
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
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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

3
Use of the boundary layer approximation and the
mixing length hypothesis in turbulence models

• Example temporal planar boundary layer

• Laminar
• Turbulent ( )
• Mixing length hypothesis
  where
• Quasisteady assumption
• Lateral momentum flux is independent
  of in some range
• Then constant, giving
• Limitations
• Describes mean flow
• Parameterization of is flow dependent
• Advection is modeled as diffusion, affecting the
  realism of couplings to microphysics

4
The mixing length concept is not inherently
restricted to average behavior
Typical eddy at height
This picture motivates a time-resolved 1D
model involving a sequence of fluid displacements
5
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)
• Resolve small eddies to obtain realistic
  advective-diffusive coupling, e.g.,
• viscous wall layer
• diffusive interfaces in buoyant flows
• thin flames in turbulent combustion

6
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.,
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
that emulates 3D eddy kinematics
The triplet map captures compressive strain and
rotational folding effects, and causes no
property discontinuities
The triplet map is implemented numerically as
a permutation of fluid cells
This procedure emulates the effect of a 3D
eddy on property profiles along a line of sight
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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
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u
v
15
w
c
10
Triplet map is applied to all properties (velociti
es u, v, w and scalar, c)
u, v, w, c
5
0
eddy range
-5
-10
0
5
10
15
20
y
Dynamic step affects velocities but not scalar
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Eddy selection procedure

• Assign a time scale t to each possible eddy,
  parameterized by eddy size and location within
  the computational domain
• The set of t values defines an eddy rate
  distribution l from which eddies are sampled
• The physics is in the determination of t
• (as in mixing length theory)

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Determination of t (schematic)

• Principle Enforce consistency of eddies and flow
  (velocity and density profiles)
• Eddy Eddy velocity so eddy energy
  ( eddy size)
• Flow gravitational potential
  energy change caused by eddy
• maximum kinetic
  energy extractable by adding wavelets
• to velocity components

Relation determining t
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 (u1 gt u2)
• Free stream density ratio s r2/r1
• 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
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Interpretationhigh density contrast inhibits
layer growth
y
r1
unrestricted entrainment
rM
restricted entrainment
r2
r

• Shear resides in the mixed layer
• Energy of entraining eddies scales as rM
• For r2 gtgt rM, the energy requirement for
  entrainment of fluid 2

restricts heavy-side intrusion
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Implied density profile for r2 gtgt r1
r1
y
smooth transition
rrms
sharp interface
peak due to flapping of sharp density gradient
r2
r
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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
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A zonal picture suggests an interpretation of
asymmetric s dependence of the growth rate
u1
light
uM
mixed
dM
u2
heavy
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
T2 T1
Imposed temperature difference T1 gt T2
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.
mean temperature
mean density

• 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
Pr 2750
Pr 0.025
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Computed midplane density fluctuations match Pr
0.7 data (curve) and predict Pr trend
Pr 0.025
Pr 0.7
Pr 2750
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Scaled ODT centerline density PDFs for Ra 1.4
10 8 expt. (Chicago) for Ra 9 1011
Scaled PDFs (ODT and measured) are insensitive to
Ra. ODT Pr dependence has not yet been tested
experimentally.
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Interpretation of ODT results Two mechanisms
contribute to core density fluctuations
Mechanism Regime of Implied
scalings Dominance Nu
rrms Stirring of mean density gradient low
Pr nonclassical classical by core
eddies Entrainment of unmixed wall-layer
high Pr classical nonclassical fluid
into core
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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)

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A slow-diffusing stable species can cause
layering of a convection process
double-diffusive instability
?
?
?
T
S
cold, fresh
solid initial state dash later state
(salt diffusivity assumed negligible)
warm, salty
?
?
??
?
S
T
S
T
convective layer forms, then heat diffuses
across stable interface, initiating a new
layer
?
?
?
?
?
S
T
S
T
thermohaline staircase
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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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ODT captures the observed regimes of diffusive
interface structure
curves measurements symbols computations for
two flow configurations
Flux ratio (salinity/temperature) across
interface
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
DY
an eddy event here is not allowed
LES/ODT overlap region
L
DY
all eddy events must extend into the inner region
no-slip wall
inner region
DY
DZ
DX
ODT sub-control volumes imbedded in an inner
region LES control volume
Illustration of the allowable locations of ODT
eddy events

• 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

42
LES channel simulation using ODT near-wall
momentum closure for Ret 1200
ODT
LES
LES
LES
LES
recent small eddy events
streamwise velocity, u
time average profile
instantaneous profile A
sharp
instantaneous profile B
smooth
y/h (distance from centerline)
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Comparison of DNS (Moser et al.) and LES/ODT
mean velocity profiles
Ret 590 LES/ODT
Ret 1200 LES/ODT
Ret 2400 LES/ODT
Ret 590 DNS
u
y
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Friction law
Solid line experimental correlation (Dean)
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Velocity fluctuations for Ret 590
Filled symbols ODT (v and w statistics are
identical) Open symbols LES/ODT Curves 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)

heated wall
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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

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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