From Navier-Stokes to Saint-Venant

Transcription

1 From Navier-Stokes to Saint-Venant Course 1 E. Godlewski 1 & J. Sainte-Marie 1 1 ANGE team LJLL - January 2017

2 Euler Eulerian vs. Lagrangian description Lagrange x(t M(t = y(t, z(t flow of the trajectories u(x, y, z, t dm dt ϕ t : R 3 R 3 M(t 0 M(t Relation Lagrangian/Eulerian description = u(m(t, t Df Dt = d f f (M(t, t = dt t + u. f. Fluid mechanics vs. solid mechanics particle position time constant

3 The Navier-Stokes equations Equations + w z = 0, ( ρ 0 t + u + w z ( w ρ 0 t + u w + w w z Role of the pressure Boundary conditions + p Completed with an energy equality = σ xx + σ xz z, + p z = ρ 0g + σ zx + σ zz z,

4 The Navier-Stokes equations (cont d Kinematic oundary conditions at the ottom z u w = 0 ( z u + z y v w = 0 at the free surface η t + η u s w s = 0 Dynamic oundary conditions (σ pi d n s = p a (x, tn s, t (n (σ pi d n = κu

5 The Navier-Stokes equations (cont d Equations + w z = 0, t + u + w z + p = σ xx + σ xz z, w t + u w + w w z + p z = g + σ zx + σ zz z, Newtonian fluids σ xx = 2µ, σ zz = 2µ w z, σ xz = µ ( z + w, σ zx = µ ( z + w,

6 Equations The Euler system + w z = 0, t + u + w z + p = 0, w t + u w + w w z + p z = g, Boundary conditions kinematic (ottom + free surface, dynamical (p s = p a Energy equality: a constraint with t ˆ η z E dz + ˆ η z u (E + p dz = 0 E = u2 + w 2 + gz 2 The Euler system and physical solutions?

7 Origins of the Euler/NS system Mass within a volume V m = V ρdv Mass conservation dm dt = ρ V t dv + ρu.ds = 0 S Green-Ostrogradsky formula ρu.ds = div (ρudv local mass conservation equation When ρ = cst S ρ t + (ρu + (ρv + (ρw = 0 y z V + v y + w z = 0

8 Origins of the Euler/NS system (cont d Divergence free condition + v y + w z = 0 Variation of velocity du = u(x + udt, y + vdt, z + wdt u(x, y, z, t i.e. du = udt + y vdt + wdt + z t dt Acceleration a defined y du = adt Fundamental law of dynamics with σ T = pi d + σ a = t + u + v y + w z ρa div (σ T = ρg

9 Models for compressile fluids Euler equation (compressile gas dynamics ρ + (ρu (ρu + (ρu2 t (ρw t E (E + p + t + (ρw z + (ρuw = 0, + (ρuw z + (ρw 2 + z w(e + p z + p = 0, + p z = 0, = 0, with p = (γ 1ρe (for polytropic gas 1 γ 3 and E = 1 2 ρ(u2 + w 2 + ρe, Compressile incompressile : singular limit

10 Free surface and compressile models We (often consider incompressile fluids ut ecause of the free surface, the models have compressile features Several velocities

11 Fluids with complex rheology Newtonian fluids σ v,xx = 2µ, σ v,zz = 2µ w z, The Mohr-Coulom criterion σ v,xz = µ ( z + w, σ v,zx = µ ( z + w, σ T = σ N tan(φ + c c: cohesion, φ: internal friction angle The Drucker-Prager criterion { σ = σv + κ σv σ v if σ v 0, σ κ else with κ = 2λ[p] + Also Herschel-Bulkley fluid,...

12 The Navier-Stokes equations Equations + w z = 0, t + u + w z + p = σ xx + σ xz z, w t + u w + w w z + p z = g + σ zx σ xx = 2µ, σ xz = σ zx = µ ( z + w Kinematic oundary conditions at the ottom z u w = 0 + σ zz z,, σzz = 2µ w z, ( z u + z y v w = 0 at the free surface η t + η u s w s = 0

13 Boundary conditions for Navier-Stokes Normals n s = ( η 2 ( η 1, n = ( z 2 ( z 1 Free surface ( µ z + w s η ( 2µ s p s = 0, s 2µ w z p s µ η ( s z + w s = 0, s.

14 Boundary conditions for Navier-Stokes (cont d At the ottom ( w µ + z z ( 2µ p + z ( 2µ w z p µ z ( z + w = κu, Mainly µ z = κu +...

15 Shallow water approximation Rescaling ε = h/λ Rescaling Time : T = λ/c Velocities : W = h/t = εc, U = λ/t = C Pressure P = C 2 Variales without dimension x = x λ, z = z h, η = η h, t = t T, p = p P, ũ = u U, and w = w W. Reynolds numer, Froude numer, ottom friction ν = µ Uλ = 1 Re, g = gh U 2 = 1 Fr 2, κ = κ U,

16 Shallow water approximation (cont d Dimensionless 2D Navier-Stokes equations ũ x + w z = 0 ũ t + ũ2 ũ w + x z ε 2 ( w t Boundary conditions + p x = ( 2 ν ũ x x ũ w + x + w 2 + p z kinematic (not modified + z ( ν ũ ε 2 z z = 1 + x + z ( 2 ν w z w + ν x ( ν ũ z + ε2 ν w x

17 Shallow water approximation (cont d Boundary conditions at the free surface ( ν ũ ε z + ε 2 w s x ε η ( 2 ν ũ s x x p s = 0, s 2 ν w z p s ε ν η ( ũ s x z + ε 2 w s x = 0, s at the ottom ( ν ε 2 w ε x + ũ z ε z ( 2 ν ũ x x p +ε z ( 2 ν w x z p ν z ( ũ x z + ε 2 w x = κũ,

18 Hydrostatic Navier-Stokes system With initial variales + w z = 0 t w z A good model + p = p z = g + Simplified role of the pressure Rather complex to analyse and solve ( 2ν ( ν z + ν w + z + z ( ν z + ν w ( 2ν w z

19 Validity of the hydrostatic assumption OK for river flows, tsunami,... Questionale for short waves

20 Vertically averaged hydrostatic Euler system Still with initial variales Hydrostatic Euler system Averaged version H t + t ˆ η z ˆ η + w z = 0 t w z udz = 0, z (ˆ η u dz + p = p a + g(η z A closure relation needed p z = g z u 2 dz + ˆ η + p = 0 z p dz = p z

21 Closure relations Rescaled viscosity & friction ν = εν 0, κ = εκ 0

22 Closure relations (cont d Rescaled oundary conditions give z = O(ε, s z = O(ε, couple with gives and hence 2 u z 2 = O(ε z = O(ε u = u + O(ε

23 The Saint-Venant system Formulation H t + ( Hū = 0, (Hū + (Hū2 + g t 2 With viscosity H t + ( Hū = 0, (Hū + (Hū2 + g t 2 H 2 H 2 pa = H gh z κu. pa = H gh z + Energy alance, vertical velocity, passive tracer Friction laws u u Navier S f = κu, Manning-Strickler S f = C f, Darcy-Weisach S f = C f u u H The Saint-Venant system in 2d H 4 3 ( ū κū 4νH 1 + κ 3ν H,