Mathematical Model

Mathematical Model

Lecture developed within the project 3/DSW/4.1.2/2008

cofunded by European Social Fund (EFS)

Václav Uruba CTU Prague, AS CR

• Claude Louis Marie Henri Navier

– 1820

• Sir George Gabriel Stokes

– 1842

Navier-Stokes Equations

Navier-Stokes Equations

Nonlinearity

CONTINUITY HYPOTHESIS

1 2

i i i i

j

j i j j

Du u u p u

Dt t u x x  x x



   

    

    

1 2

i i i

j

j i j j

U U P U

U X X Re X X



       

    

Re LV

 

div ^k 0

k

u u

x

      u 

1. Shift in space 2. Shift in time

3. Galilean transformation 4. Parity

5. Rotation 6. Scaling

N-S Symmetries

Only when nonlinearity is negligible Momentum conservation Energy conservation

• Clay Mathematics Institute, Cambridge, Massachusetts

• 7 Millennium Prize Problems, May 24, 2000

1. P versus NP

2. The Hodge conjecture

3. The Poincaré conjecture - solved by G.Perelman 4. The Riemann hypothesis

5. Yang–Mills existence and mass gap

6. Navier–Stokes existence and smoothness 7. The Birch and Swinnerton-Dyer conjecture

Millennium Prize Problems

$1,000,000 each

Methods to Study Turbulence

• Statistical approach

– Statistical-kinetic approach

– Statistical-probabilistic approach

• Deterministic approach

• Fluid molecules (Saint-Venant 1877)

• Turbulent viscosity (Bussinesq 1877)

• Reynolds decomposition (1894)

• Covariances and cross-correlations (Einstein 1914)

• Mixing length (Prandtl 1925)

• Taylor hypothesis (1935)

Statistical-kinetic approach

• Stochastic process  PDF, spectra

• Kampé de Fériet (1939), Millionshchikov (1939), Kolmogorov (1941),

Obukchov (1941), Osanger (1949) Heisenberg and

Von Weizsaker (1948)

Statistical-probabilistic approach

• Individual realization

• Coherent structures (Theodorsen 1952)

• DNS method

Deterministic approach

RANS

NS

RANS

Boussinesq Prandtl

BL boundary

_tot

_t x₂

2 2

i 1 i i k

i k k

Du p u u u

Dt x  x x



   

    

  

 ^,  ^,

eff t t t

 x    x

 ^{2 3}

P p  k

2 2

i 1 i

i k

Du p u

Dt x  x





   

 

2 2

i 1 i

eff

i k

Du P u

Dt x  x





   

 

2 1

2

t mix

l u

  ^x



 ^,t ^   ^{ } ^,t

u x u x u x

• Gives quantitative results

• No decoupling between scales

• Phenomenological models needed

• The only method applicable to engineering problems (2011)

RANS

• Resolution of scales

• Space and time

• Computers: Re ≤ 1 000

• Reality: Re ≥ 1 000 000

Direct Numerical Simulation

slope -5/3 slope 2

Re

n 

 

logE 

log 1 1l0

• Resolution of LARGE scales

• Modeling of SMALL scales

Large Eddy Simulation

slope -5/3 slope 2

 

logE 

log 1 1l0