Topics in Fluid Dynamics: Classical physics and recent ... · Topics in Fluid Dynamics: Classical...

Topics in Fluid Dynamics:Classical physics and recent mathematics

Toan T. Nguyen1,2

Penn State University

Graduate Student Seminar @ PSUJan 18th, 2018

1Homepage: http://math.psu.edu/nguyen2Math blog: https://nttoan81.wordpress.com

Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 1 / 20

Fluid motion

u(x , t)

Figure : Fluid domain: Ω ⊂ R3 and unknown fluid trajectory: xt ∈ Ω (left) orunknown fluid velocity: u(x , t) ∈ R3 (right).

• Lagrangian description (left): trajectory of each fluid molecule x ∈ Ω

xt = u(xt , t), x0 = x

• Eulerian description (right): velocity field

u(x , t) ∈ R3

at each position x ∈ Ω and time t ≥ 0.

Fluid motion

u(x , t)

xt = u(xt , t), x0 = x

u(x , t) ∈ R3

at each position x ∈ Ω and time t ≥ 0.

Fluid motion

u(x , t)

xt = u(xt , t), x0 = x

u(x , t) ∈ R3

at each position x ∈ Ω and time t ≥ 0.Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 2 / 20

Fluid motion

Classical fluid dynamics:

Continuum Hypothesis: Each point in Ω corresponds to a fluidmolecule (e.g., Hilbert’s 6th open problem: continuum limit fromN-particle system3).

Continuity equation: along particle trajectory, mass remains constant:

ρ(xt , t) det(∇xxt)dx = ρ(x , 0)dx

Figure : Illustrated the Lagrangian map: x 7→ xt for each t 6= 0.

3see my lecture notes on Kinetic Theory of Gases: https://nttoan81.wordpress.comToan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 3 / 20

Fluid motion

Classical fluid dynamics:

Continuum Hypothesis: Each point in Ω corresponds to a fluidmolecule (e.g., Hilbert’s 6th open problem: continuum limit fromN-particle system3).

Continuity equation: along particle trajectory, mass remains constant:

ρ(xt , t) det(∇xxt)dx = ρ(x , 0)dx

Figure : Illustrated the Lagrangian map: x 7→ xt for each t 6= 0.

3see my lecture notes on Kinetic Theory of Gases: https://nttoan81.wordpress.comToan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 3 / 20

Fluid motion

Incompressibility:4 volume preserving flows iff

∇ · u = 0

(Exercise: ddt J = (∇ · u)J).

∂x1u1 + ∂x2u2 = 0

In particular, fluid density ρ(x , t) remains constant along the flow. Inwhat follows, ρ = 1 (continuity equation = incompressibility).

Figure : Illustrated shear flows (left) and circular flows (right), both areincompressible.

4water can be modeled by an incompressible flow, but air is compressible.Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 4 / 20

Fluid motion

Incompressibility:4 volume preserving flows iff

∇ · u = 0

(Exercise: ddt J = (∇ · u)J).

∂x1u1 + ∂x2u2 = 0

In particular, fluid density ρ(x , t) remains constant along the flow. Inwhat follows, ρ = 1 (continuity equation = incompressibility).

Figure : Illustrated shear flows (left) and circular flows (right), both areincompressible.

4water can be modeled by an incompressible flow, but air is compressible.Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 4 / 20

Fluid motion

• Momentum equation: Newton’s law: F = ma or equivalently,

Dtu = F with Dt := ∂t + u · ∇

with F being force acting on fluid parcel:

No force: F = 0. Free particles satisfy Burgers equation (nonphysical:no particle interaction):

xx1 x2

Figure : Smooth solutions blow up in finite time (see, of course, the theoryof entropy shock solutions: Bressan, Dafermos)

Fluid motion

• Momentum equation: Newton’s law: F = ma or equivalently,

Dtu = F with Dt := ∂t + u · ∇

with F being force acting on fluid parcel:

No force: F = 0. Free particles satisfy Burgers equation (nonphysical:no particle interaction):

xx1 x2

Figure : Smooth solutions blow up in finite time (see, of course, the theoryof entropy shock solutions: Bressan, Dafermos)

Euler equations

“Ideal” fluid:5 F = −∇p, pressing normally inward on the fluidsurface (called pressure gradient):

F = −∫∂O

p ~ndσ(x)O

This yields Euler equations (1757, very classical):

Dtu = −∇p∇ · u = 0

posed on Ω ⊂ R3 with u · n = 0 on ∂Ω. NOTE: 4 equations and 4unknowns: u, p.

5as opposed to viscous fluid.Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 6 / 20

Euler equations

“Ideal” fluid:5 F = −∇p, pressing normally inward on the fluidsurface (called pressure gradient):

F = −∫∂O

p ~ndσ(x)O

This yields Euler equations (1757, very classical):

Dtu = −∇p∇ · u = 0

posed on Ω ⊂ R3 with u · n = 0 on ∂Ω. NOTE: 4 equations and 4unknowns: u, p.

5as opposed to viscous fluid.Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 6 / 20

Euler equations

• Examples (stationary):

Laminar flows (shear or circular flows):

)for arbitrary U(z)

with zero pressure gradient.

Couette flow: U(z) = z

Potential flows: u = ∇φ and so φ is harmonic:

∆φ = 0

(incompressible, irrotational flows)

Streamlines of potential flows6

6Figure: internetToan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 7 / 20

Euler equations

• Examples (stationary):

Laminar flows (shear or circular flows):

)for arbitrary U(z)

with zero pressure gradient.

Couette flow: U(z) = z

Potential flows: u = ∇φ and so φ is harmonic:

∆φ = 0

(incompressible, irrotational flows)

Streamlines of potential flows6

6Figure: internetToan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 7 / 20

Euler equations

• Vorticity: to measure the rotation in fluids

ω = ∇× u

(anti-symmetric part of ∇u, recalling x = u ≈ u0 + (∇u0)x : translation,dilation, and rotation).

Note that ω = [ω, u] (the Lie bracket), or explicitly

Dtω = ω · ∇u

Theorem (Helmholtz’s vorticity law)

Vorticity moves with the flow: ω(x , t) = x t#ω0(x).(as a consequence, vortex remains a vortex).

Hint: Compute ddt (ω − x t#ω0(x)).

Euler equations

ω = ∇× u

Dtω = ω · ∇u

Euler equations

ω = ∇× u

Dtω = ω · ∇u

Euler equations

γt St

Theorem (Kelvin’s circulation theorem)

Vorticity flux through an oriented surface or circulation around an orientedcurve is invariant under the flow:

Γγ =

∮γu · ds =

∫∫Sω · dS

Hint: A direct computation.

Euler equations

3D Euler:Dtω = ω · ∇u

• Vortex stretching: ω · ∇u, which appears “quadratic” in ω, and onecould end up with d

dtω ≈ ω2 or even d

dtω ≈ ω1+ε, whose solutions blow up

in finite time. However,

• Open problem: do smooth solutions to 3D Euler actually blow up infinite time? (no, if vorticity remains bounded, Beale-Kato-Majda ’84).

• Recent mathematics and then a proof of Onsager’s conjecture ’49: Isett,De Lellis, Szekelyhidi, Buckmaster, Vicol,.... Numerical proof of finite timeblow up: Luo-Hou, Sverak,....

Euler equations

2D Euler equations

• In fact, many physical fluid flows are essentially 2D:

Atmospheric and oceanic flows

Flows subject to a strong magnetic field, rotation, or stratification.

• In 2D, vorticity is scalar and is transported by the flow:

Dtω = 0

(no vortex stretching). In particular, vorticity remains bounded, smoothsolutions remain smooth, and weak solutions with bounded vorticity areunique (Yudovich ’63).

2D Euler equations

• In fact, many physical fluid flows are essentially 2D:

Atmospheric and oceanic flows

Flows subject to a strong magnetic field, rotation, or stratification.

• In 2D, vorticity is scalar and is transported by the flow:

Dtω = 0

(no vortex stretching). In particular, vorticity remains bounded, smoothsolutions remain smooth, and weak solutions with bounded vorticity areunique (Yudovich ’63).

2D Euler equations

• An important problem: the large time dynamics of 2D Euler. Completemixing: whether ω(tj)

∗ 0 in L∞, as tj →∞? No, due to energy

conservation. However,

Conjecture (2D inverse energy cascade, Kraichnan ’67)

Unlike 3D, energy transfers to larger and larger scales (low frequencies).

Figure : Source: van Gogh and internet

2D Euler equations

• An important problem: the large time dynamics of 2D Euler. Completemixing: whether ω(tj)

∗ 0 in L∞, as tj →∞? No, due to energy

conservation. However,

Conjecture (2D inverse energy cascade, Kraichnan ’67)

Unlike 3D, energy transfers to larger and larger scales (low frequencies).

Figure : Source: van Gogh and internet

2D Euler equations

• 2D Euler steady states:u0 · ∇ω0 = 0

With u0 = ∇⊥φ0, the stream function φ0 and vorticity ω0 have parallelgradient, hence (locally) ω0 = F (φ0), yielding

∆φ0 = F (φ0)

• Major open problem: which F determines the large time dynamics ofEuler?

Theorem (Arnold ’65)

If F is strictly convex, then steady states u0 are nonlinearly stable in H1.

Hint: Find casimir functional so that u0 is a critical point.

2D Euler equations

∆φ0 = F (φ0)

2D Euler equations

∆φ0 = F (φ0)

2D Euler equations

• A delicate question: whether Arnold stability implies asymptotic stability(recalling Euler is an Hamiltonian)?

Inviscid damping: Kelvin 1887, Orr 1907

Mathematics near Couette: Masmoudi, Bedrossian, Germain ’14-

2D Euler equations

• A delicate question: whether Arnold stability implies asymptotic stability(recalling Euler is an Hamiltonian)?

Inviscid damping: Kelvin 1887, Orr 1907

Mathematics near Couette: Masmoudi, Bedrossian, Germain ’14-

2D Euler equations

• Hydrodynamic stability: Rayleigh, Kelvin, Orr, Sommerfeld,Heisenberg,....the study of spectrum of shear flows:

) U = 0

Rayleigh (1880): U(z) that has no inflection point is spectrally stable.

Figure : Great interest in the early of 20th century (aerodynamics). Source:internet

2D Euler equations

) U = 0

2D Euler equations

) U = 0

2D Navier-Stokes equations

• Viscous fluid: F = −∇p + ν∆u (Newtonian), with fluid viscosity ν > 0:

(∂t + u · ∇)u = −∇p + ν∆u

∇ · u = 0

posed on Ω ⊂ R3 with u = 0 on ∂Ω. NOTE: u, p are unknown.

• Million-dollar open problem: whether smooth solutions to 3D NavierStokes blow up in finite time. (Like 2D Euler, smooth solutions to 2DNavier Stokes remain smooth, Ladyzhenskaya ’60s).

• ......back to Hydrodynamic Stability.

(∂t + u · ∇)u = −∇p + ν∆u

∇ · u = 0

(∂t + u · ∇)u = −∇p + ν∆u

∇ · u = 0

The role of viscosity:

d’Alembert’s paradox, 1752: Zero drag exerted on a body immersedin a potential flow (as momentum equation is in the divergence form).Birds can’t fly!

L. Prandtl, 1904: the birth of the boundary layer theory (viscousforces become significant near the boundary). This gave birth ofAerodynamics.

The role of viscosity, cont’d:

Lord Rayleigh, 1880: “Viscosity may or may not destabilize the flow”

Reynolds experiment, 1885: All laminar flows become turbulent at ahigh Reynolds number:

Re :=inertial force

viscous force=

u · ∇uν∆u

ν& 104

Recent mathematics:7 Grenier (ENS Lyon)-Toan 2017-:

Confirming the viscous destabilization (linear part with Y. Guo)

Invalidating generic Prandtl’s boundary layer expansion

Disproving the Prandtl’s boundary layer Ansatz

7for more, see my blog: https://nttoan81.wordpress.comToan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 19 / 20

Perspectives: Boundary layer cascades, bifurcation theory, stability of rollwaves, fluid mixing, and much more!

yuEuler

ν12 : Prandtl’s layer

ν34 : 1st sublayer

ν58 : 2nd sublayer· · ·

ν: Kato’s layer

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