Navier-Stokes equations. Some real world problems arising ......Navier-Stokes equations. Some real...
Transcript of Navier-Stokes equations. Some real world problems arising ......Navier-Stokes equations. Some real...
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
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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.
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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....
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Applications of fluid flows
PresiónPRESIÓNPRESIÓN
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Applications of fluid flows
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Applications of fluid flowsAPERTURA DE UN PARACAÍDASCampo de velocidades y presión
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Applications of fluid flows
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Applications of fluid flows
Von Kármán’s vortex street (repeating pattern of swirling vortices)
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Applications of fluid flows
TEMPERATURA, Mach 2
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
Applications of fluid flows
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
Applications of fluid flows
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
Applications of fluid flows
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
What is a fluid? (Plain language)
Any material that is not in solid state.
Commonly, liquid or gas/plasma.
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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
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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.
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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.).
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
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.).
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
Navier-Stokes equations
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).
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Navier-Stokes equations
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).
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Navier-Stokes equations
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.
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Navier-Stokes equations
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
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Navier-Stokes equations
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
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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.).
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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
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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
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
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
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
Dimensionless equations
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
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
Dimensionless equations
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
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
Dimensionless equations
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
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
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
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
Boundary layers and turbulence
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
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).
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
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).
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
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).
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
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
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Turbulence
Streamlines in a cavity
Re = 103 Re = 106
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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.. . .
Doc-Course, IMUS, April 2nd, 2018 Navier-Stokes equations
Navier-Stokes equations
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 )
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Navier-Stokes equations
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
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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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