Navier-Stokes equations

Navier-Stokes equations

Prof. Václav Uruba

Coordinate system

x1

x3

x2

u2

u3

u1

Fluid

• Continuum

• Viscous

• Incompressible

Ma < 0,15 (0,3) Air:

U < 50 (100) m/s

Kn l

 

12 1

2

du

   dx

konst

  pv RT

p

 RT





Continuity equitation

• General:

• Incompress.:

• Hyperbolic PDF 1st order

1 D 0

Dt



 ^{   u}

0, D 0

Dt

  u   div u  0

Newton's laws of motion

1. Law of inertia

2. Change of momentum law

3. Action-reaction law

A body continues in its state of rest, or in

uniform motion in a straight line, unless acted upon by a force.

A body acted upon by a force moves in such a manner that the time rate of change of

momentum equals the force.

If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

Forces are in equilibrium: EVERYWHERE and ANYTIME

Inertial reference frame

1687

Definition: inertial force, inertial mass

 

d m

 dtv

F d

m m

 dtv  a

Forces equilibrium

• Volume:

• Surface:

• Gauss-Ostrogradski:

• Cauchy eq.:

• Constitutive form.:

Elementary volume

i 1 ik

k

Du

Dt x





 



ij

Dui

Dt 

ij 2 ij

d  s 1 2

i j ij

j i

u u

s x x

   

     

i i i

S V i

f n dS f dV x

 

 



ij P ij dij

    

Navier-Stokes eq.

Particle acc.

Convective acc.

Local acc. Pressure gradient acc.

Acc. Volumetric forces

Friction acc.

Continuity

4 scalar equations 4 scalar unknowns: u₁, u₂, u₃, p

div ^k 0

k

u u

x

      u 

1 2

i i i i

k i

k i k k

Du u u p u

u g

Dt t x x  x x



   

     

    

  ^{1 2}

D p

Dt t 



         

 u u

u u u g

   ^{, ,t p ,t}

u x x

N-S equations

• Acceleration

• Momentum

• Forces

• Mechanical energy

1 2

i i i i

k

k i k k

Du u u p u

Dt t u x x  x x



   

    

    

2

i i i i

k

k i k k

Du u u p u

Dt t u x x x x

   ^   ^   ^   ^

    

Navier-Stokes equations

• Momentum balance

• Partial differential equations

• Stationary – elliptic

• Instationary – parabolic

• 1.o. time, 2.o. space -> 1 i.c., 2 b.c.

• 4 eq., 4 unknowns

• NONLINEAR

• Not-integrable

N-S equations – solution

• Strong solution

– Existence?

– Uniqueness?

• Week solution

– Integral equitation – Variational problem

1 2

i i i i

k

k i k k

Du u u p u

Dt t u x x  x x



   

    

    

 ^,

i i

a x t





Boundary conditions

• Wall

• „no slip“ condition

• Euler eq. (inviscid)

0

u_  u_   u 0 u_

u_

1

D p

Dt ^{  } u

0, u 0 u_  _ 

Navier-Stokes equations

• Claude Louis Marie Henri Navier (fr.) 1822

• George Gabriel Stokes (ir.) 1842

• Clay Mathematics Institute of Cambridge, Massachusetts (CMI), Paris, May 24, 2000

– 7 mat. problems for 3nd millennium, 1milUSD prize

– Proof of existence, smoothness and uniqueness (i.e. stability) of solution NSE in R₃

Nonlocality N-S

• Dynamical nonlocality

– Pressure in a point is defined by the entire velocity field.

– Pressure is non-Lagrangian – nonlocality in time („memory“).

– Equitation of vorticity (pressure) – vorticity is nonlocal.

– Two-side link of velocity and vorticity fields (i.e. vorticity is not a passive quantity).

• Reynolds decomposition

– Mutual link between the fields of mean velocities and fluctuations is not localized in time and space – character of a functional.

– Fluctuations in a given point in space and time are functions of the mean field in the entire space.

Group theory

• Definition

– Projection g(.)

– Negation, additivity, equivalency

• Group of symmetry

– A physical quantity conservation

• Group of symmetry N-S: G

– Holds:

 ^{,t N S, g G : g}   ^{N S}

u x    u 

N-S Symmetry

1. Shift in space 2. Shift in time

3. Galilean transformation 4. Mirroring (parity)

5. Rotation 6. Scaling

Cylinder in cross-flow

? ?

Cylinder in cross-flow

N-S Symmetry

• Shift in space

Momentum conservation

N-S Symmetry

• Shift in time

Zachování energie

N-S Symmetry

• Galilean tr.

N-S Symmetry

• Mirroring (parity)

N-S Symmetry

• Rotation

N-S Symmetry

• Scaling

N-S rice

• N-S equations rearranging:

– Eq. for pressure: div(N-S) – Eq. for vorticity: rot(N-S)

N-S for pressure

• Poisson eq.

• Neumann b.c.

2 k l

l k

u u p  ^{ }x x

  

 

2 2

un

p

n  ^n

 

 

N-S for vorticity

• Vorticity definition

• N-S

• EE (inviscid)

• ER stationary

„For stationary plane flow of ideal fluid in potential force field the vorticity is conserved along all streamlines.“

   rot

ω u u

  ²  

D

Dt  ^t      

 ω ω

uω ω ω u

D

Dtω  

ω u

 ^u ^{  0}

Transformation of N-S

variables Relevant quantities p-theorem

Dimensionless quantities

N-S eq.

1 parameter

, , ,

i i

u p x t , , , L V  

2 2

, , ,

i i

i i

x u p t

X U P

L V V L

 

    

2

i Re i i

k

k i k k

U U P U

U X X X X



 

   

    

       Re LV

  1 2

i i i i

k

k i k k

Du u u p u

Dt t u x x  x x



   

    

    

N-S eq. For compressible flow

Mechanical pressure

Thermodynamic pressure

2nd viscosity (volumetric)

  ^{1 ˆ 2 1}  

p 3 D

Dt t 

 

             

 u u

u u u u g

pˆ    p  u

1 D 0

Dt



 ^{   u}