Turbulence I:

Mark RastLaboratory for Atmospheric and Space PhysicsDepartment of Astrophysical and Planetary SciencesUniversity of Colorado, Boulder

ISIMA 20109 July 2010

What is the turbulence problem?Properties of turbulent flowsThe questions being asked

Tools: correlations, distributions, spectra

Conceptual underpinnings.

Turbulence is like beauty, hard to define, but you know it when you see it.

Turbulence is like humor, hard to describe why you are laughing.

Turbulence is like plumbing, who cares about the details, get the job done.

Turbulence (3 fundemental properties: random, multi-scale, vortical):

1. The Navier-Stokes equations are deterministic. Turbulent flows are unpredictable.• Velocity field varies significantly and irregularly in both position and time

• Deterministic chaos – hypersensitivity to initial conditions

• Nonlinear dynamical system with very large number of degrees of freedom

• Importance of nonlinearity depends on Reynolds number

∂u∂t

+ u ⋅∇u = −1

ρ∇P +

1

Re∇2u Re ≡

ULν

Correlation length (integral scale):

Rij (r) ≡ ui (x+ r) uj (x)

l0 =L11 =1

R11(0)R11

0

∞

∫ (r) dr R11(0) = u12

R11(r)

R11(0)

rL11

Two-point correlation(instantaneous)

Dissipation length (Kolmogorov scale):

ε ≡ energy dissipation rate (erg/g/s = cm2 /s3)

ν ≡ kinematic viscosity ((g cm−1s−1)(g−1cm3) = cm2 s−1)η ≡

ν 3

ε

⎛

⎝⎜⎞

⎠⎟

14

Kolmogorov hypothesis: ε :

u03

l0

Energy dissipation at small scales =Energy injection at integral scale

⇒

l0

η: Re3 4 Fully resolved 3D simulation

requires minimumgrid points per integral scale

N : Re9 4

Strictly random flow has degrees of freedom.Reduced by correlations within flow (coherent structures).

N : Re9 4

∇⋅u = 0

2. Multi-scale• Significant energy over many scales – scale separation an idealization

• Large scale flow depends on geometry and external forcing

• Small scale flow – “universal character”

• Period – Scale – Energy relationship: small scales, high frequency, low energylarge scales, low frequency, high energy

Re ≡ULν

=u⋅∇u

∇2u

transfer of energy from large to small scalesdissipation of small scale inhomgeneities

Werne, Fritts, et al. (1999+)DNS Kelvin-Helmholz instability

∇2u :

∂%u∂t

+ 2ν k2 %u = 0 %u(k, t) =%u(k,0)e−νk2t

Exponential damping with rate fastest at smallest scales.

u⋅∇u : eik1 ⋅xeik2 ⋅x =ei(k1 +k2 )⋅x =eik⋅x

k = k1 + k2Do triad interactions produce on average ?k > k1 and k2

Sum of randomly oriented unit vectors in three-dimensions:

1

4

3

4

• Energy transferred in both directions, but preferentially to smaller scales

3. Vortical• Turbulence is characterized by high degree of fluctuating vorticity

• Kinetic energy density greatest at large scales, Enstrophy at small scales

∂u∂t

+ u ⋅∇u = −1

ρ∇P +

1

Re∇2u

∇⋅u = 0

ω =∇×u

∂ω∂t

+ u ⋅∇ω =ω ⋅∇u +1

Re∇2ω

ω x

∂ux

∂x

ω y

∂ux

∂y

ω z

∂ux

∂z

Tilting: exchange between components

Stretching: amplification,volume of fluid element constant, radius decreases

Conservation of angular momentum (per unit mass): ω xr

2 = constantDΓDt

=0 Γ = ω∫ ⋅dA

Mininni et al. (2006)DNS with Taylor-Green forcing

Vorticity production

Energy transfer to small scales ⇒

Dω 2

Dt=2ω iω j

∂ui

∂xj

+ν∂2ω 2

∂xj∂xj

−2ν∂ω i

∂xj

∂ω i

∂xj

As much destretching as stretching?

Mean enstrophy increases, or maintains itself against viscosity only if

ω iω j

∂ui

∂x j

> 0

On average, there must be a correlation between the magnitude of the vorticity and the gradient of the velocity in the flow. This requires a velocity derivative skewness (NonGaussianity in the derivative statistics).

Heuristic argument:• random walk – on average the distance between any two points increases • infinitesimal line elements are on average stretched by random motions (and folded).

k−53

k13

Goto and Kida (2007)DNS with spectral forcing Turbulent flows

characterized by macroscopic mixing to small scales, and thus enhanced microscopic diffusion and dissipation.

Two components:

Transport – fluid parcels, “mixing” on continuum scale

Dissipation – homogenization within/between parcels, transfer to the molecular scale

So what is the Question?

What is the turbulence problem?

Is there one, or are all flows fundamentally different?

High Level:“The problem of turbulence is reduced here [for incompressible flows] to finding the probability distribution P(dω) in the phase space of turbulent flow Ω ={ω}, the points ω of which are all possible solenoidal vector fields u(x,t) which satisfy the equations of fluid mechanics and the boundary conditions imposed at the boundaries of the flow.” The fluid mechanical fields are constrained random fields and every actual example of such a field is a realization of the statistical ensemble of all possible fields. “Thus, in a turbulent flow, the equations of fluid dynamics will determine uniquely the evolution in time of the probability distribution of all the fluid dynamic fields.” – Monin and Yaglom (1971)

−1

ρ∇2P =

∂ui

∂x j

∂u j

∂xi

= eijeij − γ ijγ ij = eijeij −1

2ω i

2

∇⋅u = 0∂u∂t

+ u ⋅∇u = −1

ρ∇P +

1

Re∇2u

⇒

eij =12

∂ui

∂xj

+∂uj

∂xi

⎛

⎝⎜

⎞

⎠⎟ γij =

1

2

∂ui

∂x j

−∂u j

∂xi

⎛

⎝⎜

⎞

⎠⎟Splat Spin

⇒ • Describe the probability distribution in terms of its moments• Truncate the moment expansion to some low order (tractability)• Model the influence of higher order moments (closure)• Evolve low order model

Lower Level Description (more explicit low order model):

The problem of turbulence is the problem of scalar and vector transport at unresolved scales. Numerical solution of the Navier-Stokes equations can yield a robust realization of the flow at resolved scales, modeling of turbulent transport is needed to bridge the gap between resolved motions and molecular diffusion.

What are the characteristics of small-scale turbulent motions, how do these depend on the properties of the large-scale motions from which they derive, and knowing them, how can we model the transport of scalar and vector quantities, such as concentration, energy, or momentum?

⇒ • Numerical simulation of Navier-Stokes at affordable resolution• Subgrid model of turbulent transport below grid scales

Chapman – Enskog: • properly takes into account nonequilibrium particle distribution functions • due to the presence of the background variations – transport occurs because (distribution is nonMaxwellian ) f ≠ f0

Molecular transport (Maxwell 1866): Random molecular motions yield

– viscosity by the transport of net momentum– conductivity by the transport of net energy– diffusion by the transport of molecular identity

N+ = Φ(uz0

∞

∫ ) uzduz

N− = Φ(uz−∞

0

∫ ) uzduz

f = f0

Φ(uz) =m

2πkT⎛⎝⎜

⎞⎠⎟

1/2

exp−muz

2

2kT⎛

⎝⎜⎞

⎠⎟

Assume (lowest order):

⇓N+ =N− =

14

N u z0

z0 +23λ

z0 −23λ

ρ+ ≈1

4N u m ux (z0 ) −

2

3λ

∂ux

∂z 0

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥

ρ− ≈1

4N u m ux (z0 ) +

2

3λ

∂ux

∂z 0

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥

⇓⇐Fρ =

13

Nm u λ∂ux

∂z 0μ =

1

3Nm u λ ⇐

Dynamic Viscosity

Transport of x-momentum:

The turbulence problem: How to determine the velocity fluctuations from knowledge of the large scale flow (the closure problem).How to mix – provided by elastic collisions in molecular transport (understanding the interface between continuum and molecular dynamics)

Prandtl’s answer (as quoted by Bradshaw 1974)• typical values of the fluctuating velocity components are each proportional to

where l is the mixing length (Mischungsweg)• l “may be considered as the diameter of the masses of fluid moving as a whole in

each individual case; or again, as the distance traversed by a mass of this type before it becomes blended in with neighboring masses”

• l is “somewhat similar, as regards effect, to the mean free path in the kinetic theory of gases”

l ∂U ∂z

Mixing length theory of turbulent transport (Prandtl 1925):

μ =1

3Nm u λ

μT = ρl2 ∂U

∂z

Shear stress:

τ =ρ l∂U

∂z⎛⎝⎜

⎞⎠⎟

2Turbulent viscosity:

Two meanings of the mixing length

1. If turbulent motions were strictly random, turbulent viscosity would be proportional the average magnitude of the velocity perturbations

– It is the coherent motions that are key, phase relations are critical

2. Turbulent transport requires a “mixing length” – turbulent mean free path– Must understand the interface between continuum and molecular dynamics to understand where in the flow the transported quantity is deposited

Formulate a statistical description of turbulent coherent structures, Lagrangian dynamics in their presence, and mixing between parcels

Tools to describe hydrodynamic fields treated as continuous random variables

• probability density functions – distributions of values in occurrence• spatial correlations – distribution of values in physical space• power spectra – distribution of values in spectral space

Probability density function:

• Hydrodynamic fields are described by a joint probability density function for all (x,t)• Moments completely determine distribution• Navier-Stokes provide constraints on evolution of PDFs

P{u <ux(x,t) <u+du} =p(u)du with moments: uxn = unp(u)du

−∞

∞

∫

Mean: u =0 for u=U − U , central moments

Variance: σ 2 = u2

Skewness: S = u3 u2( )3/2

Flatness: F = u4 u2( )2 (Kurtosis K =F −3)

Gaussian distribution:

1. All moments can be expressed in terms of mean and variance

2. Central Limit Theorem:Let X1, X2, X3, …, Xn be a sequence of n independent and identically distributed random variables with finite mean μ and variance σ2. As the sample size increases the distribution of the sample average of these random variables approaches the normal distribution with mean μ and variance σ2/n, independent of the underlying common distribution. The sum of a large number of identically distributed independent variables has a Gaussian probability density, regardless of the shape of the pdf of the variables themselves.

p(U ) =1

2πσ 2

⎛⎝⎜

⎞⎠⎟

1/2

exp−U −μ( )2

2σ 2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

In general probability density functions of turbulent fields are nonGaussian• velocity generally slightly subGaussian • velocity difference (increment) significantly superGaussian (intermitancy) – elevated extremes

Mordant, Leveque, & Pinton (2004)

Mininni et al. (2008+): Forced turbulence (Taylor-Green) at resolution of 10243

F = f0

sin(kf x)cos(kf y)cos(kfz)

−cos(kf x)sin(kf y)cos(kfz)

0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Forcing scale 2π/kf , kf =2

k−53

The treatment of turbulent motions statistically, via random variables, means that the flow can only be considered as a statistical ensemble of realizations characterized as by some jont probability density for the values of all fluid dynamic variables.

The moments are the expectation values (e.g. ensemble means or probability means) of the flow over many realizations.

In practice, averaging over multiple realizations is replaced by spatial or temporal averaging (the ergodic theorem).

What are these PDFs, what should they be?What are they supposed to represent?

P{u1 <ux(M1) <u1 +du1} =pM1(u1)du1 M1 ≡(x,t)1

P{u1 <ux(M1) <u1 +du1,u2 <ux(M2 ) <u2 +du2} =pM1M2(u1,u2 )du1du2

Not independent random variables, coupled by Navier–Stokes

P{u1 <ux(M1) <u1 +du1,u2 <ux(M2 ) <u2 +du2 ,...,un <ux(Mn) <un +dun} =pM1M2 ...Mn

(u1,u2 ,...,un)du1du2 ... dun

For stationary random process and time-averaging (or homogeneous random field and spatial-averaging)Necessary and sufficient condition forquadratic convergence to the ensemble mean U

limT → ∞

uT −U 2 =0 where uT (t) =1T

u(t+τ )−T /2

T /2

∫ dτ

is that the two-point correlation (autocovariance) satisfies:buu (τ ) =[u(t+τ ] −U ][u(t)−U ]

limT → ∞

1T

buu0

T

∫ (τ ) dτ =0

The integral time scale (correlation time):

T1 =1

buu(0)buu

0

∞

∫ (τ ) dτ

ExperimentNumber

Ergodicity:

Velocity fluctuations de-correlate in time (or space).

Ergodic if integral time scale is finite, use averaging time T >T1.

buu (τ )

buu (0)

τT1

uT −U 2 ≈2T1

Tbuu(0)

if its statistical properties (such as its mean and variance) can be deduced from a single, sufficiently long sample (realization) of the process

Moment equations (Navier-Stokes constraints on evolution of distributions):

Lowest order (the Reynolds average equations):

Decompose fields into sum of their mean and fluctuations: ui =ui + ′ui

∂ ui + ′ui( )

∂t+ u j + ′u j( )

∂ ui + ′ui( )

∂x j

= −1

ρ

∂ P + ′P( )

∂xi

+ ν∂2 ui + ′ui( )

∂x j∂x j

Take the Average of each term (averages of fluctuations vanish, ensemble averaging commutes with differentiation):

∂ui

∂t+ u j

∂ui

∂x j

+∂( ′ui ′u j )

∂x j

= −1

ρ

∂P

∂xi

+ ν∂2ui

∂x j∂x j

∂ui

∂t+ u j

∂ui

∂x j

= −1

ρ

∂P

∂xi

+∂

∂x j

ν∂ui

∂x j

− ( ′ui ′u j )⎛

⎝⎜

⎞

⎠⎟

Reynolds stresses link orders(the closure problem)

Reynolds stresses act like viscosity

∂ u j + ′u j( )

∂x j

= 0

∂u j

∂x j

= 0(uiu j ) ≡ Reynolds stresses

Note: uiu j = ui + ′ui( ) uj + ′uj( ) =uiuj +uj ′ui +ui ′uj + ′ui ′uj =uiuj + ′ui ′uj

Note: ui = ui

Closure problem:

′ui = ui − ui Full Navier-Stokes minus Reynolds average equation yields:

∂ ′ui

∂t+

∂

∂xk

ui ′uk + uk ′ui + ′ui ′uk − ′ui ′uk( ) = −1

ρ

∂ ′P

∂xi

+ ν∂2 ′ui

∂xk∂xk

Multiplying by and averaging to obtain an equation for (e.g. Hinze 1959),

an evolution equation for the Reynolds stresses, yields new unknowns:

′u j

∂∂t

′ui ′u j

none of which can be expressed directly in terms of Reynolds stresses.

• third-order central moments• multiples by ν of second-order moments and their spatial derivatives• third and second moments of pressure and velocity products

′ui ′u j ′uk

M.C. Esher, Snakes, 1969 woodcut

The equations are not truncate-able or perhaps tractable:

At each step the difference between the number of unknowns and the number of equations increases.

The Friedman-Keller equations (Keller and Friedmann 1924) are an infinite chain of coupled moment equations.

Modeling the Reynolds stresses:

ρ∂ui

∂t+ ρu j

∂ui

∂x j

= −∂P

∂xi

+∂

∂x j

μ∂ui

∂x j

− ρ( ′ui ′u j )⎛

⎝⎜

⎞

⎠⎟

Divergence of momentum flux by molecular motionsand velocity fluctuations (mixing – moving and averaging)

Momentum change in volume due to flow:

&M =− ρu

A∫ u⋅dA( )

Mean i th component:

&M i =− ρ uiuj + ′ui ′uj( )A∫ njdA=− ρ

∂∂xj

uiuj + ′ui ′uj( )V∫ dV

Similarly the scalar transport equation:

∂φ∂t

= −∇ ⋅ uφ( ) + Γ∇2φ can be written in terms of mean and fluctuating components

∂φ∂t

+ u ⋅∇φ = ∇ ⋅ Γ∇φ − ′u ′φ( ) suggesting an enhanced diffusive flux

Advection

Down gradient diffusion model (turbulence mimics molecular transport):

Scalar turbulent diffusivity, so that:

− ′u ′φ = Γ T∇φ ⇒ ∂φ

∂t+ u ⋅∇φ = ∇ ⋅ Γ eff∇φ( )

∂φ∂t

+ u ⋅∇φ = ∇ ⋅ Γ ∇φ − ′u ′φ( ) with

Γeff (x,t) = Γ + Γ T(x,t)

Turbulent transport analogous to Fick’s law of molecular diffusion.

Molecular transport of momentum (surface forces on parcel of a Newtonian fluid):

τ ij = −Pδ ij + σ ijStress tensor has both static isotropic (independent of surface orientation) and deviatoric (due to fluid motion) components

σ ij = μ∂ui

∂x j

+∂u j

∂xi

⎛

⎝⎜

⎞

⎠⎟ + λ∇ ⋅uδ ij

Stokes assumption (only translational degrees of freedom – monatomic ideal gas)

Linear relationship between stress and strain rate.

λ =2

3μ

τ ij = − P +2

3μ∇ ⋅u⎛

⎝⎜⎞⎠⎟

δ ij + μ∂ui

∂x j

+∂u j

∂xi

⎛

⎝⎜

⎞

⎠⎟

Down gradient momentum flux

′ui ′uiNormal stresses (twice the turbulent kinetic energy k)

′ui ′u j Shear stresses

−ρ ′ui ′u j = −2

3ρkδ ij + ρν T

∂ui

∂x j

+∂u j

∂xi

⎛

⎝⎜

⎞

⎠⎟

∂ui

∂t+ u j

∂ui

∂x j

= −1

ρ

∂

∂xi

P +2

3ρk

⎛⎝⎜

⎞⎠⎟

+∂

∂x j

ν eff

∂ui

∂x j

+∂u j

∂xi

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Incompressible Navier-Stokes with “eddy viscosity” and modified mean pressure.

Specify νT solves closure problem (second-order closure model), eliptical problem for pressure solves for total pressure not gas.

νeff (x, t) = ν + ν T(x, t)

Turbulent transport by analogy:

Scalar turbulent viscosity (Boussinesq 1877), so that deviatoric Reynolds stress tensor is proportional to the mean rate of strain:

τ ij = − P +2

3μ∇ ⋅u⎛

⎝⎜⎞⎠⎟

δ ij + μ∂ui

∂x j

+∂u j

∂xi

⎛

⎝⎜

⎞

⎠⎟

ρ∂ui

∂t+ ρu j

∂ui

∂x j

= −∂P

∂xi

+∂

∂x j

μ∂ui

∂x j

− ρ( ′ui ′u j )⎛

⎝⎜

⎞

⎠⎟

Isotropic component(turbulent pressure)

deviatoric component(turbulent viscosity)

Note: −1

ρ∇2P =

∂ui

∂x j

∂u j

∂xi

+∂2 ( ′ui ′u j )

∂xi∂x j

=∂ui

∂x j

∂u j

∂xi

Turbulence must be vortical for turbulent shear stresses to exist:

ω i = ε ijk

∂uk

∂x j

ω =0 ⇒ ∂ui

∂x j

=∂u j

∂xi

for i ≠ j

Consider random incompressible irrotational flow: ω =ω = ′ω =0 ⇒ ∂ ′ui

∂x j

−∂ ′u j

∂xi

= 0

′ui

∂ ′ui

∂x j

−∂ ′u j

∂xi

⎛

⎝⎜

⎞

⎠⎟ =

∂

∂x j

1

2′ui ′u i

⎛⎝⎜

⎞⎠⎟

−∂

∂xi

′ui ′u j = 0 ⇒ ∂

∂xi

′ui ′u j =∂k

∂x j

(Corrsin and Kistler 1954)

In an irrotational flow, shear stresses have same effect as isotropic stresses – can be absorbed into a modified pressure and have no effect on mean flow.

In a simple shear flow (mean gradients perpendicular to mean flow), the assumed Reynolds stress / mean gradient relationship becomes a definition of the turbulent-viscosity:

−ρ ′ui ′u j = −1

3ρ ′ui ′uiδ ij + ρν T

∂ui

∂x j

+∂u j

∂xi

⎛

⎝⎜

⎞

⎠⎟

⇒ ′u1 ′u2 = −ν T

∂u1

∂x2Single covariance related to a single mean gradient.

Turbulence does not adjust rapidly enough to imposed mean straining – no general basis for local stress / rate of strain relationship.

local stress / rate of strain relationship:

• Non-local transport processes invalidate the hypothesized relationship between turbulent Reynolds stress and the local mean gradients

• Model works best when production and dissipation of turbulent kinetic energy are approximately in balance, P ε ≈1

P ε ≈0

P ε ≈0 P ε >1

Reynolds stress anisotropies produced in contraction (extensive axial and compressive lateral strain) are advected downstream and influence the mean flow in a way not described by mean rate of strain. Reynolds stresses reflect history not local mean rates of strain.

Axisymmetric contraction experiment:

Kinetic theory:

ν ≈1

3u λ

λ ≡mean free path

u ≡ mean molecular speed

Turbulence decay rate:

collision time scale τν =λu

Simple shear flow timescale: τ S ≈ LU

τ ≈kε

τν

τ S

= 1 10−10( )

ττ S

> 1 10 +( )

P ε >1

Statistical state of molecular motion adjusts rapidly to imposed shearing

Statistical state of turbulent motions adjusts slowly to imposed shearing

The turbulent viscosity is the product of a velocity and a length: νT(x, t) = u*(x, t) l*(x, t)

I. Mixing-length model: l* =lm u* =lm∂u∂y

(simple shear flow)

Clearly not correct for decaying grid turbulence, centerline of round jet, etc. where mean velocity gradient is zero but the turbulent velocity scale is not.

II. Kolmogorov-Prandtl model:

l* =lm u* =ck1/2

⇒ ν T = lm2 ∂u

∂y

⇒ ν T = c lm k1/2

Need: k1/2

Must solve model transport equation for k(x,t) – called a one-equation model:

Models of the turbulent viscosity (νT):

Strategy:• specify mixing length• solve model equation for k•

• turbulent viscosity hypothesis

• Solve RANS for

νT = c lm k1/2

′ui ′u j =2

3kδ ij − ν T

∂ui

∂x j

+∂u j

∂xi

⎛

⎝⎜

⎞

⎠⎟

u(x, t) and P(x, t) actually for P +23ρk⎛

⎝⎜⎞⎠⎟

Equation for k (turbulent kinetic energy ):

∂k

∂t+ u ⋅∇k = −∇ ⋅ ′T +P − ε

Flux: ′Ti ≡1

2′ui ′u j ′u j +

1

ρ′ui ′P − ν ′u j

∂ ′ui

∂x j

+∂ ′u j

∂xi

⎛

⎝⎜

⎞

⎠⎟

Production: P ≡− ′ui ′u j

∂ui

∂x j

Dissipation: ε ≡1

2ν

∂ ′ui

∂x j

+∂ ′u j

∂xi

⎛

⎝⎜

⎞

⎠⎟

∂ ′ui

∂x j

+∂ ′u j

∂xi

⎛

⎝⎜

⎞

⎠⎟

1

2′ui ′u i

Closure Approximations

ε =CDk 3/2

lm

′T = −ν T

σ T

∇k

where σ T is the turbulent Prandtl number.

Down gradient flux of turbulent kinetic energy

Turbulent viscosity hypothesis

Note: Close form, BUT fundamentally incomplete – mixing length lm must be specified.

∂ ′ui

∂t+

∂

∂xk

ui ′uk + uk ′ui + ′ui ′uk − ′ui ′uk( ) = −1

ρ

∂ ′P

∂xi

+ ν∂2 ′ui

∂xk∂xk

′ui ′u j =2

3kδ ij − ν T

∂ui

∂x j

+∂u j

∂xi

⎛

⎝⎜

⎞

⎠⎟

Kolmogorov hypothesis: ε :

u03

l0

Energy dissipation at small scales =Energy injection at integral scale

III. k – ε model (two-equation model):

• model transport equations for two turbulence quantities (in this case k and ε)• can be complete (constants, but flow dependent quantities (like lm) are not required).

i. Model transport equation for k (same as that in one-equation model)ii. Model transport equation for ε (empirical, RANS equation for ε describes

processes in dissipative range)

iii. Specify turbulent viscosity as (depends only on turbulence

quantities, independent of mean flow properties)

νT =Cμ k2

ε

∂ε∂t

+ u ⋅∇ε = ∇ ⋅ν T

σ ε

∇ε⎛

⎝⎜⎞

⎠⎟+ Cε1

P ε

k− Cε 2

ε 2

k

Cμ =0.09

Cε1 =1.44

Cε 2 =1.92

σ T = 1.0

σ ε =1.3

Standard values!(Launder and Sharma 1974)

• the simplest complete turbulence model• incorporated into many commercial CFD codes• has been applied to a wide range of incompressible flows and, modified, to a more limited range of compressible problems• inaccuracies stem from the underlying turbulent viscosity hypothesis and from the ε

equation• ε equation and constants can be obtained via renormalization group methods

Beyond RANS models:

• The mean velocity and Reynolds stresses are only the first and second moments of the Eulerian velocity PDF .

• PDF methods evolve an exact, but unclosed, equation for the evolution of the one point, one time Eulerian PDF of the velocity , derived from incompressible Navier–Stokes.

PDF methods:

f u(x,t);x,t( )

Pope, S.B. (2008), Turbulent FlowsDopozo, C. (1994), Recent developments in PDF methods, in Turbulent Reacting FlowsO’Brien, E.E. (1979), The probability density function (pdf) approach to reacting turbulent flows, in Topics in Applied Physics, Vol. 44

Spatial correlations and the velocity power spectrum:

Two point correlation (two-point, one-time covariance):

Rij (%

r,%x, t) ≡ ui (%

x,t) uj (%x+

%r,t)

For homogeneous turbulence: Rij is independent of

%x

Rij (%

r,%x,t)→ Rij (%

r,t)

Fourier transform: Φij (%

k,t) = e−i%k⋅

%r Rij (%

r,t) d%r∫

And its inverse:

Rij (%

r, t) =1

(2π )3ei

%k⋅%rΦij (%

k,t) d%k∫

For : %r =0

Rij (0, t) = uiuj = Φij∫ (

%k,t) d

%k

1

2Rii (0, t) =

12

uiui =12

Φii∫ (%k,t) d

%k

Define:

E(k)dk =12∫ Φii∫ (

%k,t) d

%k

Autocorrelation theorem: the power spectrum of a function is the Fourier transform of its autocorrelation

Φii (%k) = %ui (%

k)2

E(k) =

12

Φii∫ (%k,t) δ(

%k −k) d

%k

Integrate over shell: spectral kinetic energy density per unit mass – contribution to kinetic energy by modes with wavenumbers between k and k+dk

Kolmogorov (1941):

E(k ) dk ≡ energy/massin interval dk (cm2 /s2 )

ε ≡ energy/mass/second dissipation=injection(erg/g/s = cm2 /s3 )

E(k) : ε akb : ε 2 / 3k−5 / 3

local energy flux

E(k)

ηk kfk

k−5/3

• Inertial range: energy neither injected or dissipated. In a steady state, energy at any size scale depends only on injection/dissipation rate and size scale, not viscosity -- spectral slope by dimensional analysis

• Energy is injected a the integral scales: energy injection rate

ε :

u03

l0

• Disipative scales: Energy is removed at the same rate it is injected

η ≡ν 3

ε

⎛

⎝⎜⎞

⎠⎟

14

Formulate a statistical description of turbulent coherent structures, Lagrangian dynamics in their presence, and mixing between parcels

Spectra: all phase information lost model of transport in spectral space

• Fluid instabilities produces ever smaller scales from large scale motions Big whirls have little whirls, which feed on velocity; and little whirls have lesser whirls, and

so on to viscosity (in the molecular sense). (Richardson 1922 after Jonathan Swift)

Transport in physical space?