Uniqueness of Solutions to the Stochastic Observations - PowerPoint PPT Presentation

Turbulence Birnir Uniqueness of Solutions to the Stochastic Observations Navier-Stokes, the Invariant Measure and of Turbulence The Mean and Kolmogorov’s Theory the Invariant Measure The Role of Noise Björn Birnir Existence Theory Existence of Center for Complex and Non-Linear Science the Invariant Measure and Department of Mathematics, UC Santa Barbara Kolomogarov’s Theory Summary Turbulence and Finance, Sandbjerg 2008

Outline Turbulence Birnir 1 Observations of Turbulence Observations of Turbulence The Mean and the Invariant Measure 2 The Mean and the Invariant Measure The Role of Noise 3 The Role of Noise Existence Theory Existence 4 Theory Existence of Existence of the Invariant Measure 5 the Invariant Measure Kolomogarov’s 6 Kolomogarov’s Theory Theory Summary 7 Summary

Outline Turbulence Birnir 1 Observations of Turbulence Observations of Turbulence The Mean and the Invariant Measure 2 The Mean and the Invariant Measure The Role of Noise 3 The Role of Noise Existence Theory Existence 4 Theory Existence of Existence of the Invariant Measure 5 the Invariant Measure Kolomogarov’s 6 Kolomogarov’s Theory Theory Summary 7 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 "Observe the motion of the surface of the water, which Kolomogarov’s resembles that of hair, which has two motions, of which one Theory Summary 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 Observations of Turbulence The Mean and the Invariant Measure 2 The Mean and the Invariant Measure The Role of Noise 3 The Role of Noise Existence Theory Existence 4 Theory Existence of Existence of the Invariant Measure 5 the Invariant Measure Kolomogarov’s 6 Kolomogarov’s Theory Theory Summary 7 Summary

The Mean of the Solution to the Navier-Stokes Equations Turbulence Birnir Observations The flow satisfies the Navier-Stokes Equation of Turbulence The Mean and u t + u · ∇ u = ν ∆ u + ∇{ ∆ − 1 [ trace ( ∇ u ) 2 ] } the Invariant Measure The Role of However, in turbulence the fluid flow is not deterministic Noise Existence Instead we want a statistical theory Theory In applications the flow satisfies the Reynolds Averaged Existence of the Invariant Navier-Stokes Equation (RANS) Measure Kolomogarov’s Theory u t + u · ∇ u = ν ∆ u − ∇ p + ( u · ∇ u − u · ∇ u ) Summary

An Invariant Measure Turbulence What does the bar u mean? Birnir In experiments and simulations it is an ensamble Observations of Turbulence average The Mean and u = � u � the Invariant Measure Mathematically speaking the mean is an expectation, if The Role of Noise φ is any bounded function on H Existence Theory � Existence of E ( φ ( u )) = φ ( u ) d µ ( u ) (1) the Invariant H Measure Kolomogarov’s The mean is defined by the invariant measure µ on the Theory function space H where u lives Summary We must prove the existence of a unique invariant measure to make mathematical sense of the mean u

Turbulence around Cars and Aircraft Turbulence Thermal currents and gravity waves in the atmosphere Birnir also create turbulence encountered by low-flying Observations of Turbulence aircraft The Mean and the Invariant Turbulent drag prevents the design of more Measure fuel-efficient cars, aircrafts and ships The Role of Noise Existence Theory Existence of the Invariant Measure Kolomogarov’s Theory Summary

Turbulence in Design and Disease Turbulence Birnir Turbulence is harnessed in combustion engines in cars Observations and jet engines for effective combustion and reduced of Turbulence emission of pollutants The Mean and the Invariant The flow around automobiles and downtown buildings Measure is controlled by turbulence and so is the flow in a The Role of Noise diseased artery Existence Theory Existence of the Invariant Measure Kolomogarov’s Theory Summary

Outline Turbulence Birnir 1 Observations of Turbulence Observations of Turbulence The Mean and the Invariant Measure 2 The Mean and the Invariant Measure The Role of Noise 3 The Role of Noise Existence Theory Existence 4 Theory Existence of Existence of the Invariant Measure 5 the Invariant Measure Kolomogarov’s 6 Kolomogarov’s Theory Theory Summary 7 Summary

Noise-driven Instabilities Turbulence Birnir Observations of Turbulence Let U ( x 1 ) j 1 denote the mean flow (Leonardo’s principal The Mean and current), taken to be in the x 1 direction the Invariant Measure If U ′ < 0 the flow is unstable The Role of Noise The largest wavenumbers ( k ) can grow the fastest Existence The initial value problem is ill-posed Theory Existence of There is always white noise in the system that will the Invariant Measure initiate this growth Kolomogarov’s If U is large the white noise will continue to grow Theory Summary

How does Noise get inserted? Turbulence Let U 0 j 1 denote fast mean flow in one direction Birnir The equation linearized about the initial flow Observations U = U 0 j 1 + U ′ ( x 1 , − x 2 2 , − x 3 2 ) T , where T denotes the of Turbulence transpose, and u old = U + u , becomes The Mean and the Invariant Measure     u 1 x 1 The Role of U 0 ∂ x 1 u + U ′ − u 2  + U ′ − x 2  · ∇ u Noise + u t   2 2 − u 3 − x 3 Existence Theory 2 2 c 1 / 2 � Existence of d β k + U · ∇ U = ν ∆ u + t e k (2) the Invariant k Measure k � = 0 Kolomogarov’s u ( x , 0 ) = 0 Theory Summary Each e k = e 2 π ik · x comes with its own independent Brownian motion β k t , c k << 1 are small coefficients representing small (white) noise

The Exponentially-Growing Noise Turbulence Birnir The formula for the solution of the Navier-Stokes equation Observations linearized about the initial flow U ( x ) is of Turbulence The Mean and � t the Invariant e − ( 4 νπ 2 | k | 2 + 2 π iU 0 k 1 )( t − s ) × Measure � u ( x , t ) = The Role of 0 k � = 0 Noise e − U ′ ( t − s ) Existence   0 0 Theory U ′  c 1 / 2 d β k t e k + O ( | U ′ | ) 2 ( t − s ) 0 e 0   Existence of k  the Invariant U ′ 2 ( t − s ) Measure 0 0 e Kolomogarov’s Theory if | U ′ | << 1 is small. This is clearly noise that is growing Summary exponentially in time, in the x 1 direction, if U ′ < 0

Saturation by the Nonlinearities Turbulence This growth of the noise does not continue forever Birnir The exponential growth is saturated by the nonlinear Observations terms in the Navier-Stokes equation of Turbulence The Mean and The result is large noise that now drives the turbulent the Invariant Measure fluid The Role of Noise Thus for fully developed turbulence we get the Existence Stochastic Navier-Stokes driven by the large noise Theory Existence of ( ν ∆ u − u · ∇ u + ∇{ ∆ − 1 [ trace ( ∇ u ) 2 ] } ) dt du = the Invariant Measure � h 1 / 2 d β k + t e k Kolomogarov’s k Theory k � = 0 Summary Determining how fast the coefficients h 1 / 2 decay as k k → ∞ is now a part of the problem

Turbulent Flow and Boundary Layers 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

Outline Turbulence Birnir 1 Observations of Turbulence Observations of Turbulence The Mean and the Invariant Measure 2 The Mean and the Invariant Measure The Role of Noise 3 The Role of Noise Existence Theory Existence 4 Theory Existence of Existence of the Invariant Measure 5 the Invariant Measure Kolomogarov’s 6 Kolomogarov’s Theory Theory Summary 7 Summary

The Iteration Turbulence Birnir Now we find a solution u + U of the Navier-Stokes equation by Picard iteration Observations of Turbulence Starting with u 0 as the first iterate The Mean and the Invariant Measure We use the solution of the linearized equation The Role of Noise � t h 1 / 2 e − ( 4 π 2 ν | k | 2 + 2 π iU 1 k 1 )( t − s ) d β k � Existence u o ( x , t ) = s e k ( x ) Theory k 0 k � = 0 Existence of the Invariant Measure where U 1 is now the mean flow in fully-developed turbulence Kolomogarov’s Theory � t Summary K ( t − s ) ∗ [ − u ·∇ u + ∇ ∆ − 1 ( trace ( ∇ u ) 2 )] ds u ( x , t ) = u o ( x , t )+ t o (3)