NavierStokes equations Prof Vclav Uruba Coordinate system Fluid

Navier-Stokes equations Prof. Václav Uruba

Coordinate system

Fluid • Continuum • Viscous • Incompressible Ma < 0, 15 (0, 3) Air: U < 50 (100) m/s

Continuity equitation • General: • Incompress. : • Hyperbolic PDF 1 st order

Forces equilibrium • Volume: • Surface: • Gauss-Ostrogradski: • Cauchy eq. : • Constitutive form. : Elementary volume

Navier-Stokes eq. Particle acc. Local acc. Pressure gradient acc. Acc. Volumetric forces Convective acc. Friction acc. Continuity 4 scalar equations 4 scalar unknowns: u 1, u 2, u 3, p

N-S equations • Acceleration • Momentum • Forces • Mechanical energy

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

Boundary conditions • Wall • „no slip“ condition • Euler eq. (inviscid)

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 3 nd millennium, 1 mil. USD prize – Proof of existence, smoothness and uniqueness (i. e. stability) of solution NSE in R 3

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:

N-S Symmetry 1. 2. 3. 4. 5. 6. Shift in space Shift in time Galilean transformation Mirroring (parity) Rotation 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.

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. “

Transformation of N-S variables p-theorem Dimensionless quantities Relevant quantities N-S eq. 1 parameter

N-S eq. For compressible flow Mechanical pressure Thermodynamic pressure 2 nd viscosity (volumetric)