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Navier -Stokes equation s

Navier -Stokes equation s. 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 1st order. Forces equilibrium. Elementary volume. Volume: Surface:

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Navier -Stokes equation s

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  1. Navier-Stokes equations Prof. Václav Uruba

  2. Coordinate system

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

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

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

  6. Navier-Stokes eq. Local acc. Pressure gradient acc. Particle acc. Acc. Volumetric forces Convective acc. Friction acc. Continuity 4scalar equations 4scalar unknowns: u1, u2, u3, p

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

  8. 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

  9. N-S equations – solution • Strong solution • Existence? • Uniqueness? • Week solution • Integral equitation • Variational problem

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

  11. 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

  12. 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.

  13. Group theory • Definition • Projection g(.) • Negation, additivity, equivalency • Group of symmetry • A physical quantity conservation • Group of symmetry N-S: G • Holds:

  14. N-S Symmetry • Shift in space • Shift in time • Galilean transformation • Mirroring (parity) • Rotation • Scaling

  15. Cylinder in cross-flow ? ?

  16. Cylinder in cross-flow

  17. N-S Symmetry • Shift in space Momentumconservation

  18. N-S Symmetry • Shift in time Zachování energie

  19. N-S Symmetry • Galilean tr.

  20. N-S Symmetry • Mirroring (parity)

  21. N-S Symmetry • Rotation

  22. N-S Symmetry • Scaling

  23. N-S rice • N-S equations rearranging: • Eq. for pressure: div(N-S) • Eq. for vorticity: rot(N-S)

  24. N-S for pressure • Poisson eq. • Neumann b.c.

  25. 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.“

  26. Transformation of N-S variables p-theorem Dimensionlessquantities Relevantquantities N-S eq. 1 parameter

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

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