Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
Existence uniqueness and statistical
theory of the stochastic Navier-Stokes
equation in three dimensions
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Björn Birnir
Center for Complex and Nonlinear Science
and
Department of Mathematics, UC Santa Barbara
BCAM and UPV/EHU (LEIOA) 2009
Outline
Turbulence
Birnir
1
Observations of Turbulence
2
The Mean and the Invariant Measure
3
The Role of Noise
Existence
Theory
4
Existence Theory
Existence of
the Invariant
Measure
5
Existence of the Invariant Measure
6
Kolomogarov’s Theory
7
Summary
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Kolomogarov’s
Theory
Summary
Outline
Turbulence
Birnir
1
Observations of Turbulence
2
The Mean and the Invariant Measure
3
The Role of Noise
Existence
Theory
4
Existence Theory
Existence of
the Invariant
Measure
5
Existence of the Invariant Measure
6
Kolomogarov’s Theory
7
Summary
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Kolomogarov’s
Theory
Summary
A Drawing of an Eddy
Studies of Turbulence by Leonardo da Vinci
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Leonardo’s Observations
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
"Observe the motion of the surface of the water, which
resembles that of hair, which has two motions, of which one
is caused by the weight of the hair, the other by the direction
of the curls; thus the water has eddying motions, one part of
which is due to the principal current, the other to the random
and reverse motion."
Outline
Turbulence
Birnir
1
Observations of Turbulence
2
The Mean and the Invariant Measure
3
The Role of Noise
Existence
Theory
4
Existence Theory
Existence of
the Invariant
Measure
5
Existence of the Invariant Measure
6
Kolomogarov’s Theory
7
Summary
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Kolomogarov’s
Theory
Summary
The Mean of the Solution to the Navier-Stokes
Equations
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The flow satisfies the Navier-Stokes Equation
ut + u · ∇u = ν∆u + ∇{∆−1 [trace(∇u)2 ]}
The Role of
Noise
However, in turbulence the fluid flow is not deterministic
Existence
Theory
Instead we want a statistical theory
Existence of
the Invariant
Measure
In applications the flow satisfies the Reynolds Averaged
Navier-Stokes Equation (RANS)
Kolomogarov’s
Theory
Summary
u t + u · ∇u = ν∆u − ∇p + (u · ∇u − u · ∇u)
An Invariant Measure
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
What does the bar u mean?
In experiments and simulations it is an ensamble
average
u = �u�
Mathematically speaking the mean is an expectation, if
φ is any bounded function on H
�
E(φ(u)) =
φ(u)dµ(u)
(1)
H
The mean is defined by the invariant measure µ on the
function space H where u lives
We must prove the existence of a unique invariant
measure to make mathematical sense of the mean u
Outline
Turbulence
Birnir
1
Observations of Turbulence
2
The Mean and the Invariant Measure
3
The Role of Noise
Existence
Theory
4
Existence Theory
Existence of
the Invariant
Measure
5
Existence of the Invariant Measure
6
Kolomogarov’s Theory
7
Summary
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Kolomogarov’s
Theory
Summary
Noise-driven Instabilities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Let U(x1 )j1 denote the mean flow (Leonardo’s principal
current), taken to be in the x1 direction
If U � < 0 the flow is unstable
The largest wavenumbers (k ) can grow the fastest
The initial value problem is ill-posed
There is always white noise in the system that will
initiate this growth
If U is large the white noise will continue to grow
Noise-driven Instabilities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Let U(x1 )j1 denote the mean flow (Leonardo’s principal
current), taken to be in the x1 direction
If U � < 0 the flow is unstable
The largest wavenumbers (k ) can grow the fastest
The initial value problem is ill-posed
There is always white noise in the system that will
initiate this growth
If U is large the white noise will continue to grow
Noise-driven Instabilities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Let U(x1 )j1 denote the mean flow (Leonardo’s principal
current), taken to be in the x1 direction
If U � < 0 the flow is unstable
The largest wavenumbers (k ) can grow the fastest
The initial value problem is ill-posed
There is always white noise in the system that will
initiate this growth
If U is large the white noise will continue to grow
Noise-driven Instabilities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Let U(x1 )j1 denote the mean flow (Leonardo’s principal
current), taken to be in the x1 direction
If U � < 0 the flow is unstable
The largest wavenumbers (k ) can grow the fastest
The initial value problem is ill-posed
There is always white noise in the system that will
initiate this growth
If U is large the white noise will continue to grow
Noise-driven Instabilities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Let U(x1 )j1 denote the mean flow (Leonardo’s principal
current), taken to be in the x1 direction
If U � < 0 the flow is unstable
The largest wavenumbers (k ) can grow the fastest
The initial value problem is ill-posed
There is always white noise in the system that will
initiate this growth
If U is large the white noise will continue to grow
Noise-driven Instabilities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Let U(x1 )j1 denote the mean flow (Leonardo’s principal
current), taken to be in the x1 direction
If U � < 0 the flow is unstable
The largest wavenumbers (k ) can grow the fastest
The initial value problem is ill-posed
There is always white noise in the system that will
initiate this growth
If U is large the white noise will continue to grow
Saturation by the Nonlinearities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
This growth of the noise does not continue forever
The exponential growth is saturated by the nonlinear
terms in the Navier-Stokes equation
The result is large noise that now drives the turbulent
fluid
Thus for fully developed turbulence we get the
Stochastic Navier-Stokes driven by the large noise
du = (ν∆u − u · ∇u + ∇{∆−1 [trace(∇u)2 ]})dt
� 1/2
+
hk dβtk ek
k �=0
1/2
Determining how fast the coefficients hk
k → ∞ is now a part of the problem
decay as
Saturation by the Nonlinearities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
This growth of the noise does not continue forever
The exponential growth is saturated by the nonlinear
terms in the Navier-Stokes equation
The result is large noise that now drives the turbulent
fluid
Thus for fully developed turbulence we get the
Stochastic Navier-Stokes driven by the large noise
du = (ν∆u − u · ∇u + ∇{∆−1 [trace(∇u)2 ]})dt
� 1/2
+
hk dβtk ek
k �=0
1/2
Determining how fast the coefficients hk
k → ∞ is now a part of the problem
decay as
Saturation by the Nonlinearities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
This growth of the noise does not continue forever
The exponential growth is saturated by the nonlinear
terms in the Navier-Stokes equation
The result is large noise that now drives the turbulent
fluid
Thus for fully developed turbulence we get the
Stochastic Navier-Stokes driven by the large noise
du = (ν∆u − u · ∇u + ∇{∆−1 [trace(∇u)2 ]})dt
� 1/2
+
hk dβtk ek
k �=0
1/2
Determining how fast the coefficients hk
k → ∞ is now a part of the problem
decay as
Saturation by the Nonlinearities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
This growth of the noise does not continue forever
The exponential growth is saturated by the nonlinear
terms in the Navier-Stokes equation
The result is large noise that now drives the turbulent
fluid
Thus for fully developed turbulence we get the
Stochastic Navier-Stokes driven by the large noise
du = (ν∆u − u · ∇u + ∇{∆−1 [trace(∇u)2 ]})dt
� 1/2
+
hk dβtk ek
k �=0
1/2
Determining how fast the coefficients hk
k → ∞ is now a part of the problem
decay as
Saturation by the Nonlinearities
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
This growth of the noise does not continue forever
The exponential growth is saturated by the nonlinear
terms in the Navier-Stokes equation
The result is large noise that now drives the turbulent
fluid
Thus for fully developed turbulence we get the
Stochastic Navier-Stokes driven by the large noise
du = (ν∆u − u · ∇u + ∇{∆−1 [trace(∇u)2 ]})dt
� 1/2
+
hk dβtk ek
k �=0
1/2
Determining how fast the coefficients hk
k → ∞ is now a part of the problem
decay as
Outline
Turbulence
Birnir
1
Observations of Turbulence
2
The Mean and the Invariant Measure
3
The Role of Noise
Existence
Theory
4
Existence Theory
Existence of
the Invariant
Measure
5
Existence of the Invariant Measure
6
Kolomogarov’s Theory
7
Summary
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Kolomogarov’s
Theory
Summary
Uniform flow + Rotation
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
We also need oscillations when the Fourier coefficients
�(0, k2 , k3 , t) do not depend on k1
u
To deal with these component we have to start with
some rotation with amplitude A, angular velocity Ω and
axis of rotation x1
Uniform flow + Rotation on the Torus
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
The Iteration
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Now we find a solution u + U of the Navier-Stokes
equation, with swirl U = (U1 , −A1 sin(Ω1 t), A1 cos(Ω1 t))
Starting with u0 as the first iterate in the Picard iteration
We use the solution of the linearized equation
uo (x, t) =
�
k �=0
1/2
hk
�
t
0
e−(4π
2 ν|k |2 +2πi[U k +A(k ,k )])(t−s)
1 1
2 3
dβsk ek (x)
�
where U1 k1 and A(k2 , k3 )/Ω1 = A1 k22 + k32 , determine the
�(k , t) in fully-developed turbulence, and solve
swirl of u
� t
u(x, t) = uo (x, t)+ K (t−s)∗[−u·∇u+∇∆−1 (trace(∇u)2 )]ds
to
(2)
Onsager’s Conjecture in Three Dimensions
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
L2p denotes functions with the H p Sobolev norm in L2 (Ω, P)
Theorem (1)
If
1 � (1 + (2π|k |)(11/3) )
C
E(�uo � 11 + ) ≤
hk <
2
2
6
(2π|k |)
6
(3)
ess supt∈[0,∞) E(�u�211 + )(t) < C
(4)
+
2
k �=0
where the uniform flow U1 and the amplitude A1 Ω are
sufficiently large, so that
6
holds, then the integral equation (2) has unique global
solution u(x, t) in the space C([0, ∞); L211 + )
6
The Proof
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Using the oscillatory kernal we get an estimate
�
+
�u�211 + ≤ 3
(1 + (2π|k |)(11/3) )hk |Akt |2 + ��u�411 + + D + F
6
6
k �=0
Let x = �u�211 + and solve the quadratic inequality
6
x − �x 2 ≤ a + d
with x(0) = 0. Taking the expectation gives
E(x) − �E(x 2 ) ≤ E(a)
since E(d) = 0, and solving for E(x) gives
�
1
− 1� E(a)
1
1
4�2
−�
≤
E(x) ≤
2
2
2�
2�
)−E (x)
1 + E(x
(E(x)− 1 )2
2�
What are the properties of these solutions?
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
They are stochastic processes that are continuous both
in space and in time
But they are not smooth in space, they are Hölder
continuous with exponent 13
However, there is no blow-up in finite time
Existence
Theory
Instead the solutions roughens
Existence of
the Invariant
Measure
The solutions start smooth, u(x, 0) = 0, but as the
noise gets amplified they roughen, until they have
reached the characteristic roughness χ = 13 in the
statistically stationary state
Kolomogarov’s
Theory
Summary
Neither ∇u nor ∇ × u are continuous in general
What are the properties of these solutions?
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
They are stochastic processes that are continuous both
in space and in time
But they are not smooth in space, they are Hölder
continuous with exponent 13
However, there is no blow-up in finite time
Existence
Theory
Instead the solutions roughens
Existence of
the Invariant
Measure
The solutions start smooth, u(x, 0) = 0, but as the
noise gets amplified they roughen, until they have
reached the characteristic roughness χ = 13 in the
statistically stationary state
Kolomogarov’s
Theory
Summary
Neither ∇u nor ∇ × u are continuous in general
What are the properties of these solutions?
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
They are stochastic processes that are continuous both
in space and in time
But they are not smooth in space, they are Hölder
continuous with exponent 13
The Role of
Noise
However, there is no blow-up in finite time
Existence
Theory
Instead the solutions roughens
Existence of
the Invariant
Measure
The solutions start smooth, u(x, 0) = 0, but as the
noise gets amplified they roughen, until they have
reached the characteristic roughness χ = 13 in the
statistically stationary state
Kolomogarov’s
Theory
Summary
Neither ∇u nor ∇ × u are continuous in general
What are the properties of these solutions?
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
They are stochastic processes that are continuous both
in space and in time
But they are not smooth in space, they are Hölder
continuous with exponent 13
The Role of
Noise
However, there is no blow-up in finite time
Existence
Theory
Instead the solutions roughens
Existence of
the Invariant
Measure
The solutions start smooth, u(x, 0) = 0, but as the
noise gets amplified they roughen, until they have
reached the characteristic roughness χ = 13 in the
statistically stationary state
Kolomogarov’s
Theory
Summary
Neither ∇u nor ∇ × u are continuous in general
What are the properties of these solutions?
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
They are stochastic processes that are continuous both
in space and in time
But they are not smooth in space, they are Hölder
continuous with exponent 13
The Role of
Noise
However, there is no blow-up in finite time
Existence
Theory
Instead the solutions roughens
Existence of
the Invariant
Measure
The solutions start smooth, u(x, 0) = 0, but as the
noise gets amplified they roughen, until they have
reached the characteristic roughness χ = 13 in the
statistically stationary state
Kolomogarov’s
Theory
Summary
Neither ∇u nor ∇ × u are continuous in general
What are the properties of these solutions?
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
They are stochastic processes that are continuous both
in space and in time
But they are not smooth in space, they are Hölder
continuous with exponent 13
The Role of
Noise
However, there is no blow-up in finite time
Existence
Theory
Instead the solutions roughens
Existence of
the Invariant
Measure
The solutions start smooth, u(x, 0) = 0, but as the
noise gets amplified they roughen, until they have
reached the characteristic roughness χ = 13 in the
statistically stationary state
Kolomogarov’s
Theory
Summary
Neither ∇u nor ∇ × u are continuous in general
What are the properties of these solutions?
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
They are stochastic processes that are continuous both
in space and in time
But they are not smooth in space, they are Hölder
continuous with exponent 13
The Role of
Noise
However, there is no blow-up in finite time
Existence
Theory
Instead the solutions roughens
Existence of
the Invariant
Measure
The solutions start smooth, u(x, 0) = 0, but as the
noise gets amplified they roughen, until they have
reached the characteristic roughness χ = 13 in the
statistically stationary state
Kolomogarov’s
Theory
Summary
Neither ∇u nor ∇ × u are continuous in general
Outline
Turbulence
Birnir
1
Observations of Turbulence
2
The Mean and the Invariant Measure
3
The Role of Noise
Existence
Theory
4
Existence Theory
Existence of
the Invariant
Measure
5
Existence of the Invariant Measure
6
Kolomogarov’s Theory
7
Summary
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Kolomogarov’s
Theory
Summary
A Unique Invariant Measure
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
We must prove the existence of a unique invariant
measure to prove the existence of Kolmogorov’s
statistically invariant stationary state
A measure on an infinite-dimensional space H is
invariant under the transition semigroup R(t), if
�
�
Rt
ϕ(x)dµ =
ϕ(x)dµ
H
H
Here x = u(t) is the solution of N-S and Rt is induced
by the Navier-Stokes flow
We define the invariant measure by the limit
�
φ(u)dµ(u)
lim E(φ(u(t))) =
t→∞
H
(5)
Existence and Uniqueness of the Invariant
Measure
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
This limit exist if we can show that the sequence of
associated probability measures are tight
If the N-S semigroup maps bounded function on H onto
continuous functions on H, then it is called Strongly
Feller
Irreducibility says that for an arbitrary b in a bounded
set
P(sup{t<T } �u(t) − b� < �) > 0
The Proof of the Existence and Uniqueness
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
Tightness amounts to proving the existence of a
bounded and compact invariant set for the N-S flow
Strongly Feller is a generalization of a method
developed by McKean (2002) in "Turbulence without
pressure". The variation of Navier-Stokes equation is
solved to get the estimate
|Pt (φ(u)) − Pt (φ(v ))| ≤ �φ�∞ |u − v |2
Irreducibility is essentially a problem in stochastic
control theory
Outline
Turbulence
Birnir
1
Observations of Turbulence
2
The Mean and the Invariant Measure
3
The Role of Noise
Existence
Theory
4
Existence Theory
Existence of
the Invariant
Measure
5
Existence of the Invariant Measure
6
Kolomogarov’s Theory
7
Summary
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Kolomogarov’s
Theory
Summary
Kolmogorov’s Conjecture
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
We want to prove Kolmogorov’s statistical theory of
turbulence
Kolmogorov’s statistically stationary state (3-d)
�
S2 (x, t) =
|u(x + y , t) − u(y , t)|22 dµ(u) ∼ |x|2/3
H
Theorem (2)
In three dimensions there exists a statistically stationary
state, characterized by a unique invariant measure, and
possessing the Kolmogorov scaling of the structure
functions
S2 (x) ∼ |x|2/3
Proof of Kolmogorov’s Conjecture
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
Proof.
For x small the statement follows by Hölder continuity. But
the scaling of S2 (w) can be directly estimated. Thus by the
estimate (4) in Theorem 1,
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
S2 (x) ≤ C|x · (L − x)|2/3
where C is a constants, x ∈ T3 , L is a three vector with
entries the sizes of the faces of T3 and |x| ≤| L|.
Corollary (1)
There exist solutions of the Stochastic Navier-Stokes
−
equation that are not in H 2
Can We Compute the Invariant Measure?
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
The Navier-Stokes equation is
∂u
+ u · ∇u = ∆u − ∇p + ∇{∆−1 [trace(∇u)2 ]}
∂t
(6)
Let w = Du 0 u, where u 0 is the initial condition
∂w
+ u · ∇w + w · ∇u = ∆w + 2∇{∆−1 [trace(∇u · ∇w)]}
∂t
(7)
Adding "white" noise CdB we get the stochastic PDE
dw
= [∆w − u · ∇w − w · ∇u
+ 2∇{∆−1 [trace(∇u · ∇w)]}]dt + CdB
(8)
Ito Diffusion and Girsanov’s Theorem
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
The Ito diffusion for the "Eulerian" fluid particle is
√
dXt = −u(Xt , t)dt + 2νdBt
We can write the solution of (8) without the pressure term as
� t
w(x, t) = E[f (Xt )] +
E[ek (Xt−s )]CdBs
0
By Girsanov’s theorem
w(x, t) = E[f (Bt )Mt ] +
�
0
t
E[ek (Bt−s )Mt−s ]CdBs
where Mt is the Martingale
� t
�
1 t
Mt = exp{−
u(Bs , s) · dBs −
|u(Bs , s)|2 ds}
2 0
0
(9)
The Invariant Measure
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
This implies that (7) has the invariant measure
√
dµ = lim Mt d [N (eν∆t , 2ν) ∗ N (eAt , Qt )]
t→∞
(10)
�t
where the variance Qt is Qt = 0 eAs Cds, with the operator
Aw = 2∇{∆−1 [trace(∇u · ∇w)]}]. In terms of densities the
invariant measure is
dµ = lim e{−
t→∞
Rt
1
0 u(x,s)·dx− 2
Rt
0
|u(x,s)|2 ds}
�
−
|x|2
e− 2ν
√
dx
2ν
hk ûk2
e 2νλk
�
×
d ûk
2νλk /hk
k �=0
1/2
where ûk are the Fourier coefficients of u, hk
� 2 (k )
∼ |∇u|
Outline
Turbulence
Birnir
1
Observations of Turbulence
2
The Mean and the Invariant Measure
3
The Role of Noise
Existence
Theory
4
Existence Theory
Existence of
the Invariant
Measure
5
Existence of the Invariant Measure
6
Kolomogarov’s Theory
7
Summary
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Kolomogarov’s
Theory
Summary
Conclusions
Turbulence
Birnir
Observations
of Turbulence
Starting with sufficiently fast constant flow in some
direction there exist turbulent solutions
The Mean and
the Invariant
Measure
These solutions are Hölder continuous with exponent
(Onsager’s conjecture)
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
1
3
There exist a unique invariant measure corresponding
to these solutions
This invariant measure give a statistically stationary
state where the second structure function of turbulence
scales with exponent 23 (Kolmogorov’s conjecture)
In one and two dimensions the same works but the
scaling exponents are respectively 3/2 (Hack’s Law)
and 2 (Batchelor-Kraichnan Theory)
Conclusions
Turbulence
Birnir
Observations
of Turbulence
Starting with sufficiently fast constant flow in some
direction there exist turbulent solutions
The Mean and
the Invariant
Measure
These solutions are Hölder continuous with exponent
(Onsager’s conjecture)
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
1
3
There exist a unique invariant measure corresponding
to these solutions
This invariant measure give a statistically stationary
state where the second structure function of turbulence
scales with exponent 23 (Kolmogorov’s conjecture)
In one and two dimensions the same works but the
scaling exponents are respectively 3/2 (Hack’s Law)
and 2 (Batchelor-Kraichnan Theory)
Conclusions
Turbulence
Birnir
Observations
of Turbulence
Starting with sufficiently fast constant flow in some
direction there exist turbulent solutions
The Mean and
the Invariant
Measure
These solutions are Hölder continuous with exponent
(Onsager’s conjecture)
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
1
3
There exist a unique invariant measure corresponding
to these solutions
This invariant measure give a statistically stationary
state where the second structure function of turbulence
scales with exponent 23 (Kolmogorov’s conjecture)
In one and two dimensions the same works but the
scaling exponents are respectively 3/2 (Hack’s Law)
and 2 (Batchelor-Kraichnan Theory)
Conclusions
Turbulence
Birnir
Observations
of Turbulence
Starting with sufficiently fast constant flow in some
direction there exist turbulent solutions
The Mean and
the Invariant
Measure
These solutions are Hölder continuous with exponent
(Onsager’s conjecture)
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
1
3
There exist a unique invariant measure corresponding
to these solutions
This invariant measure give a statistically stationary
state where the second structure function of turbulence
scales with exponent 23 (Kolmogorov’s conjecture)
In one and two dimensions the same works but the
scaling exponents are respectively 3/2 (Hack’s Law)
and 2 (Batchelor-Kraichnan Theory)
Conclusions
Turbulence
Birnir
Observations
of Turbulence
Starting with sufficiently fast constant flow in some
direction there exist turbulent solutions
The Mean and
the Invariant
Measure
These solutions are Hölder continuous with exponent
(Onsager’s conjecture)
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
1
3
There exist a unique invariant measure corresponding
to these solutions
This invariant measure give a statistically stationary
state where the second structure function of turbulence
scales with exponent 23 (Kolmogorov’s conjecture)
In one and two dimensions the same works but the
scaling exponents are respectively 3/2 (Hack’s Law)
and 2 (Batchelor-Kraichnan Theory)
Conclusions
Turbulence
Birnir
Observations
of Turbulence
Starting with sufficiently fast constant flow in some
direction there exist turbulent solutions
The Mean and
the Invariant
Measure
These solutions are Hölder continuous with exponent
(Onsager’s conjecture)
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
1
3
There exist a unique invariant measure corresponding
to these solutions
This invariant measure give a statistically stationary
state where the second structure function of turbulence
scales with exponent 23 (Kolmogorov’s conjecture)
In one and two dimensions the same works but the
scaling exponents are respectively 3/2 (Hack’s Law)
and 2 (Batchelor-Kraichnan Theory)
Applications
Turbulence
Birnir
Observations
of Turbulence
Better closure approximations for RANS
The Mean and
the Invariant
Measure
There is no universal way now of approximating the
eddy viscosity (closure problem)
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary
u · ∇u − u · ∇u ∼ νeddy |∇u|(∇u + ∇T u)
Knowing µ we will be able to solve the closure problem
to any desired degree of accuracy
The same applies to LES (Large Eddy Simulations)
Better subgrid models for LES
The Artist by the Water’s Edge
Leonardo da Vinci Observing Turbulence
Turbulence
Birnir
Observations
of Turbulence
The Mean and
the Invariant
Measure
The Role of
Noise
Existence
Theory
Existence of
the Invariant
Measure
Kolomogarov’s
Theory
Summary