Navier-Stokes equations. Some real world problems arising ......Navier-Stokes equations. Some real world problems arising in Fluid Mechanics Francisco Ortegón Gallego Departamento

Navier-Stokes equations.Some real world problems arising in Fluid Mechanics

Francisco Ortegón GallegoDepartamento de Matemáticas,

Universidad de Cádiz, Spain

Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations

Navier-Stokes equations are a general system of PDEsgoverning all fluid flows.

ContinuumMechanics

•Fluid

Mechanics

– Newtonian Fluids

– Non-Newtonian Fluids

• SolidMechanics

– Plastic . . . . . . . . . . . . . . . .

– Elastic

Rheology


What is fluid mechanics?

Robert A. Granger, Fluid Mechanics, Dover, 1995:

Fluid mechanics is the branch of physics sciences concernedwith how fluids behave at rest or in motion. Its uses arelimitless. We must understand fluid mechanics if we wantto model the red spot on Jupiter, or measure the vorticity ina tornado, or design a transonic wing for an SST(supersonic transport), or predict the behavior of subatomicparticles in a betatron. To track the motions of fluids pastobjects or through objects, in oceans or in molecules, here onearth or in distant galaxies, fluid mechanics examines thebehavior of liquids, gases, and plasma—of everything thatis not solid.


Applications of fluid flows

Aerodynamics and Hydrodynamics.Combustion and Propulsion.Meteorology.Oceanography.Hydrology.Civil Engineering.Ecology and Environmental Science.Magnetohydrodynamics.Medical Science: blood flow.Petroleum Industry.Traffic flow....



PresiónPRESIÓNPRESIÓN




Applications of fluid flowsAPERTURA DE UN PARACAÍDASCampo de velocidades y presión





Von Kármán’s vortex street (repeating pattern of swirling vortices)



TEMPERATURA, Mach 2








What is a fluid? (Plain language)

Any material that is not in solid state.

Commonly, liquid or gas/plasma.


What is a fluid? (mechanical approach)

A fluid is a material that does not support shear stress.Shear stress is stress that is applied parallel or tangential to theface of a material.That is the reason why fluids take the shape of their containers!

Area ADisplacement d

Velocity u = dtimeForce f

height h

Stress fA

Strain dh

Strain rate uh


What is a fluid? (mathematical definition)

Definition

A continuum is a fluid if(i) it is isotropic, and(ii) the stress tensor is a function of the rate strain tensor

(σ = F (∇u+∇uT )).

Isotropic means that the mechanical properties are notdependent on the direction along which they are measured:D = ∇u+∇uT then

σ = F (D)⇐⇒ σ′ = F (D′)

where D′ = QDQT and σ′ = QσQT for any orthogonal matrixQ with detQ = 1.


Navier-Stokes equations

Fluid Mechanics is based upon five great principles ofphysics:

1 conservation of mass;2 conservation of linear momentum (Newton’s second law);3 conservation of angular momentum;4 conservation of energy; and5 second law of thermodynamics.

The first four are used for both fluid mechanics and solidmechanics.The fifth principle is only used in fluid mechanics in the case ofa gas. It is then used in order to deduce a state law (that is, anequation relating some physics unknowns, as pressure, density,temperature, etc.).



Fluid Mechanics is based upon five great principles ofphysics:

1 conservation of mass;2 conservation of linear momentum (Newton’s second law);3 conservation of angular momentum;4 conservation of energy; and5 second law of thermodynamics.

The first four are used for both fluid mechanics and solidmechanics.The fifth principle is only used in fluid mechanics in the case ofa gas. It is then used in order to deduce a state law (that is, anequation relating some physics unknowns, as pressure, density,temperature, etc.).



Fluid flows are governed by a system of PDEs that is deducedfrom: (1) the principles above, and (2) the definition of a fluid.

ρ,t +∇ · (ρu) = 0

(ρu),t +∇ · (ρu⊗ u)−∇ · τ +∇p = ρf(ρ

(|u|2

2+ e

)),t

+∇ ·[ρ

(|u|2

2+ e

)+ p

]u

= ∇ · (τu)−∇ · q + ρfu

τ is the viscous stress tensor, (σ = −pI + τ )q is the heat flux (e. g. q = −k∇e, k > 0).



From (i) and (ii) in the definition of the fluid, it can be shownthat the viscous stress tensor has the following structure

τ = f0I + f1D + f2D2, D =

1

2(∇u+∇uT ).

If τ depends on∇u+∇uT in a linear way (Newtonian fluids),then

τ = λ∇ · u I + µ(∇u+∇uT )

where λ and µ are the Lamé coefficients of viscosity (they maydepend on the density and the temperature).

λ = −2µ/N (Stokes’ law).



For an incompressible fluid (∇ · u = 0) we obtainρ,t +∇ · (uρ) = 0

ρ (u,t + (u · ∇)u)− µ∆u+∇p = ρf

∇ · u = 0

This system is called incompressible variable densityNavier-Stokes equations.

Notice that, in this situation, these equations are not coupledwith the energy equation.



If moreover the fluid is homogeneous (ρ = ρ(t)) then

ρ,t +∇ · (uρ) = ρ,t + (∇ · u)ρ+ u∇ρ = ρ,t = 0,

and thus the density does not depend on t either: ρ = ρ0.Putting ν = µ/ρ0 (kinematic viscosity) and p instead of p/ρ0(kinematic pressure) we obtain the so-called (transient)incompressible, homogenous, viscous and NewtonianNavier-Stokes equations, namely

u,t + (u · ∇)u− ν∆u+∇p = f

∇ · u = 0



C. L. M. H. Navier George Gabriel Stokesb. February 10, 1785 b. August 13, 1819

Dijon, France Skreen, County Sligo, Irland†August 21, 1836 †February 1, 1903

Paris, France Cambridge, Cambridgeshire, England


Mathematical problem

u,t + (u · ∇)u− ν∆u+∇p = f in Ω× (0, T )

∇ · u = 0 in Ω× (0, T )

u|t=0 = u0 in Ω

u = 0 on ∂Ω× (0, T )

u,t local acceleration;(u · ∇)u convective acceleration;

u,t + (u · ∇)u acceleration along streamlines;−ν∆u viscous term (friction deceleration).∇p pressure acceleration (direct and normal action of

the medium);f external forces (gravity, Coriolis, electromagnetics,

etc.).


Dimensionless equations

Let L be a characteristic length of the domain Ω,U a characteristic velocity,T = L/U.u = u/U, p = p/U2, x = x/L, t = t/TThus, u, p, x and t are dimensionless quantities.

u,t + (u · ∇)u+∇p− νLU∆u = f

∇ · u = 0

u|t=0 = u0

u = 0

where f = fL/U2, u0 = u0/U

Re =LU

νis the so-called Reynolds number




u,t + (u · ∇)u+∇p− νLU∆u = f

∇ · u = 0

u|t=0 = u0

u = 0


Re =LU





u,t + (u · ∇)u+∇p− νLU∆u = f

∇ · u = 0

u|t=0 = u0

u = 0


Re =LU




When L=O(1), U=O(1) =⇒ Re =1

νu,t + (u · ∇)u− ν∆u+∇p = f in Ω× (0, T )

∇ · u = 0 in Ω× (0, T )

u|t=0 = u0 in Ω

u = 0 on ∂Ω× (0, T )

Some partial known results on the existence and/or uniqueness of solutions:J. Leray (1933,1934), E. Hopf (1951), J. L. Lions, G. Prodi (1959), J. Serrin(1963), O. A. Ladyzhenskaya (1969), J. L. Lions (1969), C. Foias, G. Prodi(1981),. . .But the complete mathematical resolution of this problem is still open!!!The complete resolution of the Navier-Stokes equations has been declared bythe Clay Mathematics Institute as one of the seven Millennium Prize Problemswith $1 million allocated to the solution.http://www.claymath.org/millennium-problems/navier?stokes-equation


http://www.claymath.org/millennium-problems/navier?stokes-equation



When L=O(1), U=O(1) =⇒ Re =1

νu,t + (u · ∇)u− ν∆u+∇p = f in Ω× (0, T )

∇ · u = 0 in Ω× (0, T )

u|t=0 = u0 in Ω

u = 0 on ∂Ω× (0, T )






When L=O(1), U=O(1) =⇒ Re =1

νu,t + (u · ∇)u− ν∆u+∇p = f in Ω× (0, T )

∇ · u = 0 in Ω× (0, T )

u|t=0 = u0 in Ω

u = 0 on ∂Ω× (0, T )





Boundary layers and turbulence

For high Reynolds numbers (ν → 0) two type of phenomenamay occur, namely:

1 Boundary layers.2 Turbulence.

When ν → 0, inertial effects dominate viscous effects|(u · ∇)u| |ν∆u| (?).

v

δ δ ∼= Re−1/2

u = 0

u


Boundary layers and turbulence


Turbulence

When ν → 0, the solution of the Navier-Stokes equationsbecomes more and more fluctuating in time and space.

Turbulence: chaotic state of the velocity field of a fluid flow asits Reynolds number becomes increasingly higher.

Indeed, the smallest fluctuations of u occur at a distance nearRe−3/4.Consequently, an adequate numerical simulation wouldrequire a triangulation of Ω with size h = O

(ν3/4

)at least.

⇒ The current computers may solve the NS equations up toRe < 10000 (by a direct simulation).


Turbulence




(ν3/4

)at least.



Turbulence




(ν3/4

)at least.



Turbulence

U L ν Re

spermatozoon 10−4 10−4 10−6 10−2

kite 1 1 0.15× 10−4 6.67× 104

automobile 22 3 0.15× 10−4 4.40× 106

passanger plane 250 60 0.15× 10−4 109

supersonic plane 600 20 0.15× 10−4 8.00× 108

submarine 11 50 10−6 5.50× 108


Turbulence

Streamlines in a cavity

Re = 103 Re = 106


Turbulence

A few examples of turbulent flows:the boundary layers in the atmosphere,the ocean currents,the solar photosphere, star photosphere, etc.,the wake of a reactor,the boundary layers around plane wings,the trail of ships, cars, airplanes, submarine, etc.,the flow on a river, etc.. . .



Evolution Navier-Stokes equations

Let Ω ⊂ RN , T > 0 and f : Ω× (0, T ) 7→ RN be given.

Unknowns: u : Ω× (0, T ) 7→ RN (velocity field) andp : Ω× (0, T ) 7→ R (pressure)

u,t + (u · ∇)u− ν∆u+∇p = f in Ω× (0, T )

∇ · u = 0 in Ω× (0, T )

+ initial condition in Ω

+ boundary conditions on ∂Ω× (0, T )



Steady-state Navier-Stokes equations

Let Ω ⊂ RN and f : Ω 7→ RN be given.

Unknowns: u : Ω 7→ RN (velocity field) and p : Ω 7→ R (pressure)(u · ∇)u− ν∆u+∇p = f in Ω

∇ · u = 0 in Ω

+ boundary conditions on ∂Ω


H. Brezis, Analyse fonctionnelle. Théorie et applications. Masson, Paris,1983.

G. Duvaut, Mécanique des milieux continus. Masson, Paris, 1990.

V. Girault, P. A. Raviart, Finite Element Methods for Navier-StokesEquations. Theory and Algorithms. Springer series in computationalmathematics, 5. Springer, Berlin, 1986.

R. A. Granger, Fluid Mechanics. Dover Publications, New York, 1995.

P. L. Lions, Mathematical Topics in Fluid Mechanics. Volume 1:Incompressible Models. Oxford Lecture Series in Mathematics and itsApplications, 3. Oxford Science Publications, 1996.

F. Ortegón Gallego, Algunos problemas matemáticos de la mecánica defluidos. Actas del Encuentro de Matemáticos Andaluces. Volumen 1:Conferencias plenarias y semblanzas. Emilio Briales et al. Eds.Universidad de Sevilla, Colección Abierta, nº52, pp. 185-210. Sevilla,2001.

R. Temam, Navier-Stokes equations. Studies in Mathematics and itsApplicacions, 2. North Holland, Amsterdam, 1979.


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Navier-Stokes equations. Some real world problems arising ......Navier-Stokes equations. Some real world problems arising in Fluid Mechanics Francisco Ortegón Gallego Departamento