Conservation Laws of Fluid Dynamics Models

Transcription

1 Conservation Laws of Fluid Dynamics Models Prof. Alexei Cheviakov (Alt. English spelling: Alexey Shevyakov) Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada June 2015 A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

2 Outline 1 Fluid Dynamics Equations 2 CLs of Constant-Density Euler and N-S Equations 3 CLs of Helically Invariant Flows 4 CLs of An Inviscid Model in Gas Dynamics 5 CLs of a Surfactant Flow Model 6 Discussion A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

3 Outline 1 Fluid Dynamics Equations 2 CLs of Constant-Density Euler and N-S Equations 3 CLs of Helically Invariant Flows 4 CLs of An Inviscid Model in Gas Dynamics 5 CLs of a Surfactant Flow Model 6 Discussion A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

4 Definitions Fluid/gas flow in 3D Independent variables: t, x, y, z. Dependent variables: u = (u 1, u 2, u 3 ) = (u, v, w); p; ρ. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

5 Definitions Q(x,t) Q(x + x,t) Fluid/gas flow ina 3D x x + x x direction Independent variables: t, x, y, z. Dependent variables: u = (u 1, u 2, u 3 ) = (u, v, w); p; ρ. 2D picture: y y + y n flow streamlines v n n y n B D x x x + x A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

6 Main Equations of Gas/Fluid Flow Euler equations: ρ t + (ρ u) = 0, ρ(u t + (u )u) + p = 0. Navier-Stokes equations (viscosity ν = const): ρ t + (ρ u) = 0, ρ(u t + (u )u) + p ν 2 u = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

7 Main Equations of Gas/Fluid Flow Euler equations: ρ t + (ρ u) = 0, ρ(u t + (u )u) + p = 0. Navier-Stokes equations (viscosity ν = const): ρ t + (ρ u) = 0, ρ(u t + (u )u) + p ν 2 u = 0. 4 equations, 5 unknowns. Closure required. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

8 Main Equations of Gas/Fluid Flow Euler equations: ρ t + (ρ u) = 0, ρ(u t + (u )u) + p = 0. Navier-Stokes equations (viscosity ν = const): ρ t + (ρ u) = 0, ρ(u t + (u )u) + p ν 2 u = 0. Closure e.g. 1, homogeneous flow (e.g., water): ρ = const, div u = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

9 Main Equations of Gas/Fluid Flow Euler equations: ρ t + (ρ u) = 0, ρ(u t + (u )u) + p = 0. Navier-Stokes equations (viscosity ν = const): ρ t + (ρ u) = 0, ρ(u t + (u )u) + p ν 2 u = 0. Closure e.g. 2, incompressible flow: div u = 0, ρ t + u ρ = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

10 Main Equations of Gas/Fluid Flow Euler equations: ρ t + (ρ u) = 0, ρ(u t + (u )u) + p = 0. Navier-Stokes equations (viscosity ν = const): ρ t + (ρ u) = 0, ρ(u t + (u )u) + p ν 2 u = 0. Other closure choices: ideal gas/adiabatic, isothermal, polytropic (gas dynamics), etc... A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

11 Main Equations of Gas/Fluid Flow Euler equations: ρ t + (ρ u) = 0, ρ(u t + (u )u) + p = 0. Navier-Stokes equations (viscosity ν = const): ρ t + (ρ u) = 0, ρ(u t + (u )u) + p ν 2 u = 0. Multiple other fluid models exist. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

12 Outline 1 Fluid Dynamics Equations 2 CLs of Constant-Density Euler and N-S Equations 3 CLs of Helically Invariant Flows 4 CLs of An Inviscid Model in Gas Dynamics 5 CLs of a Surfactant Flow Model 6 Discussion A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

13 Conservation Laws of Constant-Density Euler Equations Constant-density Euler equations: u = 0, Google Image Result for u t + (u )u + p = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

14 Conservation Laws of Constant-Density Euler Equations Constant-density Euler equations: u = 0, u t + (u )u + p = 0. CLs in a general setting. Additional CLs in a symmetric setting (e.g., axisymmetric). More additional CLs in a reduced setting (e.g., planar flow). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

15 Conservation Laws of Constant-Density Euler Equations Euler equations: u = 0, u t + (u )u + p = 0. Some conservation laws known forever, e.g., [Batchelor (2000)]. Kovalevskaya form w.r.t. x, y, z. It remains an open problem to determine the upper bound of the CL order for the Euler system. Let us seek CLs using the Direct method, 2nd-order multipliers [C., Oberlack (2014)]: Λ σ = Λ σ(45 variables); Λ σr σ Φi x i = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

16 Conservation Laws of Constant-Density Euler Equations Euler equations: u = 0, u t + (u )u + p = 0. Conservation of generalized momentum: x-direction: Multipliers: Arbitrary f (t). t (f (t)u1 ) + x + y Similar in y-, z-directions. ( ) (u 1 f (t) xf (t))u 1 + f (t)p ((u 1 f (t) xf (t))u 2 ) + z ) ((u 1 f (t) xf (t))u 3 = 0. Λ 1 = f (t)u 1 xf (t), Λ 2 = f (t), Λ 3 = Λ 4 = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

17 Conservation Laws of Constant-Density Euler Equations Euler equations: u = 0, u t + (u )u + p = 0. Conservation of angular momentum x u: x-direction: t (zu2 yu 3 ) + ) ((zu 2 yu 3 )u 1 x + ( ) (zu 2 yu 3 )u 2 + zp + ( ) (zu 2 yu 3 )u 3 yp = 0. y z Multipliers: Similar in y-, z-directions. Λ 1 = u 2 z u 3 y, Λ 2 = 0, Λ 3 = z, Λ 4 = y. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

18 Conservation Laws of Constant-Density Euler Equations Euler equations: u = 0, u t + (u )u + p = 0. Conservation of kinetic energy: x-direction: Multipliers: ( ) t K + (K + p) u = 0, K = 1 2 u 2. Λ 1 = K + p, Λ i = u i, i = 1, 2, 3. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

19 Conservation Laws of Constant-Density Euler Equations Euler equations: u = 0, u t + (u )u + p = 0. Generalized continuity equation: For arbitrary k(t): (k(t) u) = 0. Multipliers: Λ 1 = k(t), Λ 2 = Λ 3 = Λ 4 = 0. Arbitrary k(t). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

20 Conservation Laws of Constant-Density Euler Equations Euler equations: u = 0, u t + (u )u + p = 0. Conservation of helicity: Vorticity: ω = curl u. Helicity: h = u ω. Helicity conservation law: h + (u E + (ω u) u) = 0, t where E = K + p is the total energy density. Topological significance/vortex line linkage. Multipliers: Λ 1 = 0, Λ i = ω i, i = 1, 2, 3. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

21 Conservation Laws of Constant-Density Euler Equations Euler equations: u = 0, u t + (u )u + p = 0. Vorticity system: conservation of vorticity. Vorticity: ω = curl u. Vorticity equations: div ω = 0, ω t + curl (ω u) = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

22 Conservation Laws of Constant-Density Euler Equations Euler equations: u = 0, u t + (u )u + p = 0. Vorticity system: potential vorticity. Vorticity equations: div ω = 0, ω t + curl (ω u) = 0. CL: (ω F ) t + (β F F t ω) = 0, β ω u. Multipliers: Λ 1 = D tf, Λ 2 = D xf, Λ 2 = D y F, Λ 2 = D zf, holding for an arbitrary differential function F = F [u, p]. Details [Müller (1995)], generalizations: [C. & Oberlack (2014)]. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

23 Plane Euler Flows; Conservation of Enstrophy Euler classical two-component plane flow: u z = ω x = ω y = 0; z = 0. (u x ) x + (u y ) y = 0, (u x ) t + u x (u x ) x + u y (u x ) y = p x, (u y ) t + u x (u y ) x + u y (u y ) y = p y ; { ω z + (u x ) y (u y ) x = 0, (ω z ) t + u x (ω z ) x + u y (ω z ) y = 0. From Wikipedia, the free encyclopedia File:Vorticity Figure 03 c.png - Wikipedia, the free encyclopedia No higher resolution available. Vorticity_Figure_03_c.png ( pixels, file size: 10 KB, MIME type: image/ This is a file from the Wikimedia Commons. Information from its d A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

24 Plane Euler Flows; Conservation of Enstrophy Euler classical two-component plane flow: u z = ω x = ω y = 0; z = 0. (u x ) x + (u y ) y = 0, (u x ) t + u x (u x ) x + u y (u x ) y = p x, (u y ) t + u x (u y ) x + u y (u y ) y = p y ; { ω z + (u x ) y (u y ) x = 0, (ω z ) t + u x (ω z ) x + u y (ω z ) y = 0. Enstrophy Conservation Enstrophy: E = ω 2 = (ω z ) 2. Material conservation law: d dt E = Dt E + Dx (ux E) + D y (u y E) = 0. Was only known to hold for plane flows, (2 + 1)-dimensions. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

25 Plane Euler Flows; Conservation of Enstrophy Euler classical two-component plane flow: u z = ω x = ω y = 0; z = 0. (u x ) x + (u y ) y = 0, (u x ) t + u x (u x ) x + u y (u x ) y = p x, (u y ) t + u x (u y ) x + u y (u y ) y = p y ; { ω z + (u x ) y (u y ) x = 0, (ω z ) t + u x (ω z ) x + u y (ω z ) y = 0. Other Plane Flow CLs Several additional vorticity-related CLs known for plane flows (e.g., [Batchelor (2000)]); A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

26 Conservation Laws of Navier-Stokes Equations Navier-Stokes Equations equations in dimensions u = 0, u t + (u )u + p ν 2 u = 0. Vorticity formulation: u = 0, ω = u, ω t + (ω u) ν 2 ω = 0. Basic conservation laws: Momentum / generalized momentum: Θ = f (t)u i, i = 1, 2, 3. Angular momentum: Θ = (r u) i, i = 1, 2, 3. Vorticity: Θ = ω i, i = 1, 2, 3. Potential vorticity. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

27 Outline 1 Fluid Dynamics Equations 2 CLs of Constant-Density Euler and N-S Equations 3 CLs of Helically Invariant Flows 4 CLs of An Inviscid Model in Gas Dynamics 5 CLs of a Surfactant Flow Model 6 Discussion A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

28 Examples of Helical Flows in Nature Fig. 3. Axial force coefficient as function of tip-speed ratio, l; with tip pitch angle, Y; as a parameter (from [15]). Wind turbine wakes in aerodynamics [Vermeer, Sorensen & Crespo, 2003] Fig. 6. Flow visualisation experiment at TUDelft, showing two revolutions of tip vortices for a two-bladed rotor (from [24]). Fig. 4. Flow visualisation with smoke, revealing the tip vortices (from [16]). Fig. 7. Flow visualisation with smoke grenade in tip, revealing smoke trails for the NREL turbine in the NASA-Ames wind tunnel (from Hand [13]). become entwined into one. Unluckily, there are no recordings of this phenomena. During the full scale experiment of NREL at the NASA-Ames wind tunnel, also flow visualisation were performed with smoke emanated from the tip (see Fig. 7). With this kind of smoke trails, it is not clear whether the smoke trail reveals the path of the tip vortex or some streamline in the tip region. Also, these A. Cheviakov Fig. 5. (UofS, FlowCanada) visualisation with smoke, revealing smoke Conservation trails Laws experiments II have been performed at very low June thrust / 35

29 Examples of Helical Flows in Nature Helical instability of rotating viscous jets [Kubitschek & Weidman, 2007] A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

30 Examples of Helical Flows in Nature Helical water flow past a propeller A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

31 Examples of Helical Flows in Nature Wing tip vortices, in particular, on delta wings [Mitchell, Morton & Forsythe, 1997] AIAA a) b) c) d) e) Fig. 9: Detached Eddy Simulation results of the 70 delta wing at α = 27 and Rec = 1.56x10 6 for five different grids. Isosurfaces of vorticity colored by spanwise vorticity component are presented. a) Coarse Grid-1.2M cells, b) Medium Grid-2.7M cells, c) Fine Grid-6.7M cells, d) Real Fine Grid-10.7M cells, e) Adavtive Mesh Refinement Grid-3.2M cells. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

32 Helical Coordinates New conservation laws for helical flows 5 z h r e» y e x e r Figure 1. An illustration of the helix ξ = const for a = 1, b = h/2π, where h is the z step over one helical turn. Basis unit vectors in the helical coordinates. Helical Coordinates Cylindrical coordinates: (r, ϕ, z). Helical coordinates: (r, η, ξ) It should be noted that helical coordinates by (r, η, ξ) are not orthogonal. In fact, it can be shown that ξ = though az + bϕ, the coordinates η = aϕ r, b ξ z are orthogonal, there exists no third coordinate orthogonal to both r and ξ that can be consistently r, a, b = const, 2 a2 + b 2 > 0. introduced in any open ball B R 3. However, an orthogonal basis is readily constructed at any point except for the origin, as follows (see Figure 1): e r = r, e ξ = ξ, e η = η = eξ e r. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

33 Helical Coordinates New conservation laws for helical flows 5 z h r e» y e x e r Figure 1. An illustration of the helix ξ = const for a = 1, b = h/2π, where h is the z step over one helical turn. Basis unit vectors in the helical coordinates. Orthogonal Basis e It should be noted r = r r that, helical e ξ = ξ coordinates ξ, e η = η by (r, η, ξ) η are = not e ξ e orthogonal. r. In fact, it can be shown that though the coordinates r, ξ are orthogonal, there exists no third coordinate orthogonal to both r and ξ that can be consistently introduced in r any open ball B R 3. Scaling factors: H r = 1, H η = r, H ξ = B(r), B(r) = However, an orthogonal basis is readily constructed at any point a2 r 2 + except b. 2 for the origin, as follows (see Figure 1): e r = r, e ξ = ξ, e η = η = eξ e r. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

34 Helical Coordinates New conservation laws for helical flows 5 z h r e» y e x e r Figure 1. An illustration of the helix ξ = const for a = 1, b = h/2π, where h is the z step over one helical turn. Basis unit vectors in the helical coordinates. Vector expansion u = u r e r + u ϕ e ϕ + u z e z = u r e r + u η e η + u ξ e ξ. It should be noted that helical coordinates by (r, η, ξ) are not orthogonal. In fact, it can be shown that though the coordinates r, ξ are orthogonal, there exists no third coordinate u η = u e η = B (au ϕ br ) ( ) orthogonal to both r and ξ that can uz, u ξ b = u e ξ = B be consistently introduced inr any uϕ + au z. open ball B R 3. However, an orthogonal basis is readily constructed at any point except for the origin, as follows (see Figure 1): e r = r, e ξ = ξ, e η = η = eξ e r. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

35 Helical Coordinates New conservation laws for helical flows 5 z h r e» y e x e r Figure 1. An illustration of the helix ξ = const for a = 1, b = h/2π, where h is the z step over one helical turn. Basis unit vectors in the helical coordinates. Helical invariance: generalizes axal and translational invariance Helical coordinates: r, ξ = az + bϕ, η = aϕ bz/r 2. It should be noted that helical coordinates by (r, η, ξ) are not orthogonal. In fact, it can be General shown helical that though symmetry: the coordinates f = f (r, r, ξ), ξ area, orthogonal, b 0. there exists no third coordinate orthogonal to both r and ξ that can be consistently introduced in any open ball B R 3. However, Axial: a = an1, orthogonal b = 0. basis z-translational: is readily constructed a = 0, b = at 1. any point except for the origin, as follows (see Figure 1): e r = r, e ξ = ξ, e η = η = eξ e r. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

36 Helically Invariant Navier-Stokes Equations Navier-Stokes Equations: u = 0, u t + (u )u + p ν 2 u = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

37 Helically Invariant Navier-Stokes Equations Navier-Stokes Equations: u = 0, u t + (u )u + p ν 2 u = 0. Continuity: 1 r ur + (u r ) r + 1 B (uξ ) ξ = 0 A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

38 Helically Invariant Navier-Stokes Equations Navier-Stokes Equations: u = 0, u t + (u )u + p ν 2 u = 0. r-momentum: (u r ) t + u r (u r ) r + 1 ( ) 2 B uξ (u r ) ξ B2 b r r uξ + au η = p r [ 1 + ν r (r(ur ) r ) r + 1 B 2 (ur ) ξξ 1 r 2 ur 2bB (a(u η ) r 2 ξ + br )] (uξ ) ξ A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

39 Helically Invariant Navier-Stokes Equations Navier-Stokes Equations: u = 0, u t + (u )u + p ν 2 u = 0. η-momentum: (u η ) t + u r (u η ) r + 1 B uξ (u η ) ξ + a2 B 2 u r u η r [ 1 = ν r (r(uη ) r ) r + 1 B 2 (uη ) ξξ + a2 B 2 (a 2 B 2 2) u η + 2abB ( (u r ) r 2 r 2 ξ ( Bu ξ) ) ] r A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

40 Helically Invariant Navier-Stokes Equations Navier-Stokes Equations: u = 0, u t + (u )u + p ν 2 u = 0. ξ-momentum: (u ξ ) t + u r (u ξ ) r + 1 B uξ (u ξ ) ξ + 2abB2 u r u η + b2 B 2 u r u ξ = 1 r 2 r 3 B p ξ [ 1 + ν r (r(uξ ) r ) r + 1 B 2 (uξ ) ξξ + a4 B 4 1 u ξ + 2bB r 2 r ( b r 2 (ur ) ξ + ( ) )] ab r uη r A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

41 Helically Invariant Vorticity Formulation Navier-Stokes Equations, Vorticity Formulation: u = 0, u =: ω = ω r e r + ω η e η + ω ξ e ξ, ω t + (ω u) ν 2 ω = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

42 Helically Invariant Vorticity Formulation Navier-Stokes Equations, Vorticity Formulation: u = 0, u =: ω = ω r e r + ω η e η + ω ξ e ξ, ω t + (ω u) ν 2 ω = 0. Vorticity definition: ω r = 1 B (uη ) ξ, ω η = 1 B (ur ) ξ 1 ( ru ξ) 2abB2 u η + a2 B 2 u ξ, r r r 2 r ω ξ = (u η ) r + a2 B 2 r u η A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

43 Helically Invariant Vorticity Formulation Navier-Stokes Equations, Vorticity Formulation: u = 0, u =: ω = ω r e r + ω η e η + ω ξ e ξ, ω t + (ω u) ν 2 ω = 0. r-momentum: (ω r ) t + u r (ω r ) r + 1 B uξ (ω r ) ξ = ω r (u r ) r + 1 B ωξ (u r ) ξ [ 1 + ν r (r(ωr ) r ) r + 1 B 2 (ωr ) ξξ 1 r 2 ωr 2bB (a(ω η ) r 2 ξ + br )] (ωξ ) ξ A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

44 Helically Invariant Vorticity Formulation Navier-Stokes Equations, Vorticity Formulation: u = 0, u =: ω = ω r e r + ω η e η + ω ξ e ξ, ω t + (ω u) ν 2 ω = 0. η-momentum: (ω η ) t + u r (ω η ) r + 1 B uξ (ω η ) ξ a2 B 2 r [ 1 + ν (u r ω η u η ω r ) + 2abB2 (u ξ ω r u r ω ξ ) = ω r (u η ) r 2 r + 1 B ωξ (u η ) ξ r (r(ωη ) r ) r + 1 B 2 (ωη ) ξξ + a2 B 2 (a 2 B 2 2) ω η + 2abB ( (ω r ) r 2 r 2 ξ ( Bω ξ) ) ] r A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

45 Helically Invariant Vorticity Formulation Navier-Stokes Equations, Vorticity Formulation: u = 0, u =: ω = ω r e r + ω η e η + ω ξ e ξ, ω t + (ω u) ν 2 ω = 0. ξ-momentum: (ω ξ ) t + u r (ω ξ ) r + 1 B uξ (ω ξ ) ξ [ 1 + ν + 1 a2 B 2 r (u ξ ω r u r ω ξ ) = ω r (u ξ ) r + 1 B ωξ (u ξ ) ξ r (r(ωξ ) r ) r + 1 B 2 (ωξ ) ξξ + a4 B 4 1 ω ξ + 2bB r 2 r ( b r 2 (ωr ) ξ + ( ) )] ab r ωη r A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

46 Conservation Laws for Helically Symmetric Flows For helically symmetric flows: Seek local conservation laws using divergence expressions i.e., Γ 1 t + Γ2 + Γ3 r ξ = r Θ t + Φ Θ t + 1 r r (rφr ) + 1 Φ ξ B ξ = 0 [ t Θ Γ1 r, ( Γ 1 r ) + 1 r r ) (r Γ2 + 1 r B Φr Γ2 r, Φξ B r Γ3. ( )] B ξ r Γ3 = 0, 1st-order multipliers in primitive variables. 0th-order multipliers in vorticity formulation. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

47 Conservation Laws for Helically Symmetric Inviscid Flows: ν = 0 Primitive variables - EP1 - Kinetic energy Θ = K, Φ r = u r (K + p), Φ ξ = u ξ (K + p), K = 1 2 u 2. Primitive variables - EP2 - z-momentum Θ = B ( br ) uη + au ξ = u z, Φ r = u r u z, Φ ξ = u ξ u z + abp. Primitive variables - EP3 - z-angular momentum Θ = rb (au η + br ) uξ = ru ϕ, Φ r = ru r u ϕ, Φ ξ = ru ξ u ϕ + bbp. Primitive variables - EP4 - Generalized momenta/angular momenta Θ = F ( r B uη), Φ r = u r F ( r B uη), Φ ξ = u ξ F ( r B uη), where F ( ) is an arbitrary function. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

48 Conservation Laws for Helically Symmetric Inviscid Flows: ν = 0 Vorticity formulation - EV1 - Conservation of helicity Helicity: The conservation law: Θ = h, h = u ω = u r ω r + u η ω η + u ξ ω ξ. Φ r = ω r (E (u η ) 2 ( u ξ) 2 ) + u r (h u r ω r ), Φ ξ = ω ξ ( E (u r ) 2 (u η ) 2) + u ξ ( h u ξ ω ξ), where E = 1 2 u 2 + p = 1 2 ( ( (u r ) 2 + (u η ) 2 + u ξ) ) 2 + p is the total energy density. In vector notation: h + (u E + (ω u) u) = 0. t A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

49 Conservation Laws for Helically Symmetric Inviscid Flows: ν = 0 Vorticity formulation - EV2 - Generalized helicity Helicity: h = u ω = u r ω r + u η ω η + u ξ ω ξ. ( ( r [ ( r ( h H t B uη)) + H B uη) [u E + (ω u) u] + Eu η r e η H B uη)] = 0 for an arbitrary function H = H ( ). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

50 Conservation Laws for Helically Symmetric Inviscid Flows: ν = 0 Vorticity formulation - EV3 - Vorticity conservation laws where Q(t) is an arbitrary function. Θ = Q(t) ω ϕ, r Φ r = 1 ( Q(t)[u r ω ϕ ω r u ϕ ] + Q (t)u z), r Φ ξ = ab ( [ Q(t) u η ω ξ u ξ ω η] ) + Q (t)u r, r Vorticity formulation - EV4 - Vorticity conservation law ) Θ = rb (a 3 ω η b3 r 3 ωξ, Φ r Φ ξ = 2a 2 u r u z a 3 Br (u r ω η u η ω r ) + Bb3 r 2 ( u r ω ξ u ξ ω r ), = a 3 B [ (u r ) 2 + (u η ) 2 (u ξ ) 2 + r ( u η ω ξ u ξ ω η)] + 2a2 bb u η u ξ. r A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

51 Conservation Laws for Helically Symmetric Inviscid Flows: ν = 0 Vorticity formulation - EV5 - Vorticity conservation law Θ = B r 2 ( b 2 r 2 Φ r =a 3 rb B 2 ωξ + a 3 r ( 4 br )) ωη + aω ξ = Br ( b 2 r 2 2 (2u (au r η + br ) ) uξ + b (u r ω η u η ω r ) B 2 ωξ + a3 r 4 B ωz a4 r 4 + a 2 r 2 b 2 + b 4 ( ) r u r ω ξ u ξ ω r, a 2 r 2 + b ( 2 ( Φ ξ = a 3 bb (u r ) 2 + (u η ) 2 (u ξ ) 2 + r u η ω ξ u ξ ω η)) + 2a 4 rbu η u ξ. ), Vorticity formulation - EV6 - Vorticity conservation law for an arbitrary N(t, ξ). Φ = 0, Φ r = Nω r 1 B N ξu η, Φ ξ = Nω ξ, Generalization of the obvious divergence expression (G(t)ω) = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

52 Conservation Laws for Helically Symmetric Viscous Flows Primitive variables - NSP1 - z-momentum. Θ = u z, Φ r = u r u z ν(u z ) r, Φ ξ = u ξ u z + abp ν B (uz ) ξ. Primitive variables - NSP2 - generalized momentum Θ = r B uη, Φ r = r [ B ur u η ν 2aB (au η + 2 br ) ( ] r uξ + B uη) r = r [ ( B ur u η ν 2au ϕ r ] + B uη), r Φ ξ = r B uη u ξ ν 1 B [ 2abB 2 r ( u r r + B ξ] uη). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

53 Conservation Laws for Helically Symmetric Viscous Flows Vorticity formulation - NSV1 - Family of vorticity conservation laws Θ = Φ r = Φ ξ = Q(t) r 1 r B r B (aω η + br ) ωξ { Q(t) Q(t)ν = Q(t) ω ϕ, r [ u r B (aω η + br ) ωξ [ ab b 2 B r ωη + r (a 2 r 2 + b 2 ) { aq(t) [ u η ω ξ u ξ ω η] + aq (t)u r + Q(t) ν r 3 [ r 3 B ω r B (au η + br )] uξ ( aω η ξ + b ) ]} r ωξ ξ + 2brω r, (aω η + br ωξ ) + B + Q (t)b ( br ) uη + au ξ ( aω η r + b r ωξ r )]}, for an arbitrary function Q(t). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

54 Conservation Laws for Helically Symmetric Viscous Flows Vorticity formulation - NSV2 - Vorticity conservation law Θ = ) rb (a 3 ω η b3 r 3 ωξ, Φ r = B r 2 ( a 3 r 3 (u r ω η u η ω r ) b 3 ( u r ω ξ u ξ ω r )) 2a 2 Bu r ( b r uη + au ξ ) Φ ξ = B r 2 ν [ r 2 B 2 (aω η + br ) ) ωξ r (a 3 3 ω ηr b3 r 3 ωξ r + abb 2 r ( b 3 a 3 B ( (u r ) 2 + (u η ) 2 (u ξ ) 2 + r ( u η ω ξ u ξ ω η)) + 2a2 bb u η u ξ r ) + 2a2 bb ν [(1 b2 ω r + r )] (a 2 3 ηξ b3 ω r a 2 r 2 2a 2 bb r 3 ωξ ξ. r 3 ωη + a 3 ω ξ )], A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

55 Conservation Laws for Helically Symmetric Viscous Flows Vorticity formulation - NSV3 - Vorticity conservation law Θ = B r 2 ( b 2 r 2 Φ r = Φ ξ = a 3 rb B 2 ωξ + a 3 r ( 4 br )) ωη + aω ξ = Br ( b 2 r 2 2 (2u (au r η + br ) ) uξ + b (u r ω η u η ω r ) + B r 2 ( a 4 r 4 + a 2 r 2 b 2 + b 4) (ω ξ ) r + ab B B 2 ωξ + a3 r 4 B ωz a4 r 4 + a 2 r 2 b 2 + b 4 r ( u r ω ξ u ξ ω ) r a 2 r 2 + b [ 2 +ν 4a 3 B (au η + br ) uξ a 3 brb(ω η ) r + Br (b 4 a 4 r 4 a6 r 6 3 a 2 r 2 + b ( ) ] ω η, a 4 r 4 (a 2 r 2 + b 2 ) 2 ), ) ω ξ a 3 bb ( (u r ) 2 + (u η ) 2 (u ξ ) 2 + r ( u η ω ξ u ξ ω η)) + 2a 4 rbu η u ξ [ 1 ( +ν a 4 r 4 + a 2 r 2 b 2 + b 4) ] (ω ξ ) r 2 ξ a 3 br(ω η ) ξ 4a3 bb u r + 2b4 B ω r. r r 3 A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

56 Some Conservation Laws for Two-Component Flows Generalized enstrophy for inviscid plane flow (known) Θ = N(ω z ), Φ x = u x N(ω z ), Φ y = u y N(ω z ), for an arbitrary N( ), equivalent to a material conservation law d dt N(ωz ) = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

57 Some Conservation Laws for Two-Component Flows Generalized enstrophy for inviscid plane flow (known) Θ = N(ω z ), Φ x = u x N(ω z ), Φ y = u y N(ω z ), for an arbitrary N( ), equivalent to a material conservation law d dt N(ωz ) = 0. Generalized enstrophy for inviscid axisymmetric flow ( ) ( ) ( ) 1 Θ = S r ωϕ, Φ r = u r 1 S r ωϕ, Φ z = u z 1 S r ωϕ for arbitrary S( ). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

58 Some Conservation Laws for Two-Component Flows Generalized enstrophy for inviscid plane flow (known) Θ = N(ω z ), Φ x = u x N(ω z ), Φ y = u y N(ω z ), for an arbitrary N( ), equivalent to a material conservation law d dt N(ωz ) = 0. Generalized enstrophy for inviscid axisymmetric flow ( ) ( ) ( ) 1 Θ = S r ωϕ, Φ r = u r 1 S r ωϕ, Φ z = u z 1 S r ωϕ for arbitrary S( ). Several additional conservation laws arise for plane and axisymmetric, inviscid and viscous flows (details in paper). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

59 Some Conservation Laws for Two-Component Flows 18 O. Kelbin, A.F. Cheviakov, M. Oberlack, z» = const u»! r u r y x Figure 2. A schematic of a two-component helically invariant flow, with zero velocity component Generalizedin enstrophy the invariant η-direction: for general u η = inviscid 0. Conversely, helical the vorticity 2-component has only one flow nonzero component ω η 0. ( ) ( ) ( ) B Θ Note = that T the equation (6.2c) vanishes when νab = 0, i.e., for inviscid flows, and r ωη, Φ r = u r B T r ωη, Φ ξ = u ξ B T r ωη, for viscous flows with axial or planar symmetry. For other cases when the equation for an arbitrary (6.2c) T does ( ), not equivalent vanish, itto imposes a material an additional conservation differential lawconstraint on the velocity components u r, u ξ. Such a restriction may lead to lack of solution existence for boundary value problems, and hence below we( only consider ) d B the inviscid case with a, b 0 and both viscous and inviscid cases when a = 0 or b dt T r ωη = Additional conservation laws for general inviscid two-component helically invariant flows We now consider two-component helically invariant Euler flows satisfying (6.1). The A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

60 Summary for helical flows: Helically-Invariant Equations Full three-component Euler and Navier-Stokes equations written in helically-invariant form. Two-component reductions. Additional Conservation Laws Three-component Euler: Generalized momenta. Generalized helicity. Additional vorticity CLs. Three-component Navier-Stokes: Additional CLs in primitive and vorticity formulation. Two-component flows: Infinite set of enstrophy-related vorticity CLs (inviscid case). Additional CLs in viscous and inviscid case, for plane and axisymmetric flows. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

61 Outline 1 Fluid Dynamics Equations 2 CLs of Constant-Density Euler and N-S Equations 3 CLs of Helically Invariant Flows 4 CLs of An Inviscid Model in Gas Dynamics 5 CLs of a Surfactant Flow Model 6 Discussion A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

62 Conservation laws of an Inviscid Model in Gas Dynamics Euler equations: ρ t + (ρu) = 0, ρ(u t + (u )u) + p = 0. A CL classification for 2D, 3D barotropic model: p = p(ρ) (S = const). [Anco & Dar (2010)]. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

63 Outline 1 Fluid Dynamics Equations 2 CLs of Constant-Density Euler and N-S Equations 3 CLs of Helically Invariant Flows 4 CLs of An Inviscid Model in Gas Dynamics 5 CLs of a Surfactant Flow Model 6 Discussion A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

64 Surfactants - Brief Overview Surfactants Surfactant = Surface active agent. Act as detergents, wetting agents, emulsifiers, foaming agents, and dispersants. Consist of a hydrophobic group (tail) and a hydrophilic group (head). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

65 Surfactants - Brief Overview Surfactants Surfactant = Surface active agent. Act as detergents, wetting agents, emulsifiers, foaming agents, and dispersants. Consist of a hydrophobic group (tail) and a hydrophilic group (head). Sodium lauryl sulfate: A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

66 Surfactants - Brief Overview Surfactants Surfactant = Surface active agent. Act as detergents, wetting agents, emulsifiers, foaming agents, and dispersants. Consist of a hydrophobic group (tail) and a hydrophilic group (head). Hydrophilic groups (heads) can have various properties: A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

67 Surfactants - Applications Surfactant molecules adsorb at phase separation interfaces. Stabilization of growth of bubbles / droplets. Creation of emulsions of insoluble substances. Multiple industrial and medical applications. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

68 Surfactants - Applications Can form micelles, double layers, etc. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

69 Surfactants - Applications Soap bubbles... A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

70 Surfactant Transport Equations Derivation: Can be derived as a special case of multiphase flows with moving interfaces and contact lines: [Y.Wang, M. Oberlack, 2011] Illustration: Y. Wang, M. Oberlack Fig. 1 The investigated material domain B with its outer boundary B consisting of three-phase subdomains B (i) (i = 1, 2, 3) surrounded by their outer boundaries B (i) and phase interfaces S (i) A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

71 Surfactant Transport Equations (ctd.) n = 0 u Parameters Surfactant concentration c = c(x, t). Flow velocity u(x, t). Two-phase interface: phase separation surface Φ(x, t) = 0. Unit normal: n = Φ Φ. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

72 Surfactant Transport Equations (ctd.) n = 0 u Surface gradient Surface projection tensor: p ij = δ ij n i n j. Surface gradient operator: s = p = (δ ij n i n j ) x. j Surface Laplacian: s F = (δ ij n i n j ) ( (δ x j ik n i n k ) F ). x k A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

73 Surfactant Transport Equations (ctd.) n = 0 u Governing equations Incompressibility condition: u = 0. Fluid dynamics equations: Euler or Navier-Stokes. Interface transport by the flow: Φ t + u Φ = 0. Surfactant transport equation: c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

74 Surfactant Transport Equations (ctd.) n = 0 u Fully conserved form Specific numerical methods (e.g., discontinuous Galerkin) require the system to be written in a fully conserved form. Straightforward for continuity, momentum, and interface transport equations. Can the surfactant transport equation be written in the conserved form? c t + u i c u i cn x i i n j α(δ x j ij n i n j ) ( (δ x j ik n i n k ) c ) = 0. x k A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

75 CLs of the Surfactant Dynamics Equations: The Convection Case Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

76 CLs of the Surfactant Dynamics Equations: The Convection Case Governing equations (α = 0) Multiplier ansatz R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. Λ i = Λ i (t, x, Φ, c, u, Φ, c, u, 2 Φ, 2 c, 2 u). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

77 CLs of the Surfactant Dynamics Equations: The Convection Case Governing equations (α = 0) Multiplier ansatz R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. Λ i = Λ i (t, x, Φ, c, u, Φ, c, u, 2 Φ, 2 c, 2 u). Conservation Law Determining Equations E u j (Λ σ R σ ) = 0, j = 1,..., 3; E Φ (Λ σ R σ ) = 0; E c(λ σ R σ ) = 0. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

78 CLs of the Surfactant Dynamics Equations: The Convection Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. Principal Result 1 (multipliers) There exist an infinite family of multiplier sets with Λ 3 0, i.e., essentially involving c. Family of conservation laws with Λ 3 = Φ K(Φ, c Φ ). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

79 CLs of the Surfactant Dynamics Equations: The Convection Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. Principal Result 1 (divergence expressions) Usual form: Material form: G(Φ, c Φ ) + ( ) u i G(Φ, c Φ ) = 0. t x i d G(Φ, c Φ ) = 0. dt A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

80 CLs of the Surfactant Dynamics Equations: The Convection Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) x i = 0, R 3 = c t + u i c x i cn i n j u i x j = 0. Simplest conservation law with c-dependence Can take G(Φ, c Φ ) = c Φ. t (c Φ ) + ( ) u i c Φ = 0. x i A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

81 The Convection-Diffusion Case Governing equations (α 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) = 0, x i R 3 = c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

82 The Convection-Diffusion Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) = 0, x i R 3 = c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k Principal Result 2 (multipliers) Λ 1 = ΦF(Φ) Φ 1 ( x j Λ 2 = F(Φ) Φ 1 ( x j Λ 3 = F(Φ) Φ, ( c Φ ) ) 2 Φ cn x j i n j, x i x ( j c Φ ) ) 2 Φ cn x j i n j, x i x j A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

83 The Convection-Diffusion Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) = 0, x i R 3 = c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k Principal Result 2 (divergence expressions) An infinite family of conservation laws: (c F(Φ) Φ ) + ( ) A i F(Φ) Φ = 0, t x i where A i = cu i α ( (δ ik n i n k ) c ), i = 1, 2, 3, x k and F is an arbitrary sufficiently smooth function. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

84 The Convection-Diffusion Case (ctd.) Governing equations (α = 0) R 1 = ui x i = 0, R 2 = Φ t + (ui Φ) = 0, x i R 3 = c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k Simplest conservation law with c-dependence Can take F(Φ) = 1: (c Φ ) + ( ) A i Φ = 0. t x i Surfactant dynamics equations can be written in a fully conserved form. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

85 Outline 1 Fluid Dynamics Equations 2 CLs of Constant-Density Euler and N-S Equations 3 CLs of Helically Invariant Flows 4 CLs of An Inviscid Model in Gas Dynamics 5 CLs of a Surfactant Flow Model 6 Discussion A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

86 Discussion Fluid & gas dynamics: a large number of general and specific models exist. viscous and inviscid; single and multi-phase; non-newtonian; special reductions/geometries of interest; asymptotic models (KdV, shallow water, etc.). A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

87 Discussion Fluid & gas dynamics: a large number of general and specific models exist. viscous and inviscid; single and multi-phase; non-newtonian; special reductions/geometries of interest; asymptotic models (KdV, shallow water, etc.). CLs can be found systematically DCM/symbolic software. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

88 Discussion Fluid & gas dynamics: a large number of general and specific models exist. viscous and inviscid; single and multi-phase; non-newtonian; special reductions/geometries of interest; asymptotic models (KdV, shallow water, etc.). CLs can be found systematically DCM/symbolic software. General Euler and NS for 3D: basic CLs known; infinite family of potential vorticity CLs;single and multi-phase; Additional CLs for symmetric reductions; Further additional CLs for 2-component velocity. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

89 Discussion Fluid & gas dynamics: a large number of general and specific models exist. viscous and inviscid; single and multi-phase; non-newtonian; special reductions/geometries of interest; asymptotic models (KdV, shallow water, etc.). CLs can be found systematically DCM/symbolic software. General Euler and NS for 3D: basic CLs known; infinite family of potential vorticity CLs;single and multi-phase; Additional CLs for symmetric reductions; Further additional CLs for 2-component velocity. A lot remains to be discovered! A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

90 Discussion Fluid & gas dynamics: a large number of general and specific models exist. viscous and inviscid; single and multi-phase; non-newtonian; special reductions/geometries of interest; asymptotic models (KdV, shallow water, etc.). CLs can be found systematically DCM/symbolic software. General Euler and NS for 3D: basic CLs known; infinite family of potential vorticity CLs;single and multi-phase; Additional CLs for symmetric reductions; Further additional CLs for 2-component velocity. A lot remains to be discovered! Further applications: numerical simulations; development of specialized numerical methods, etc. A. Cheviakov (UofS, Canada) Conservation Laws II June / 35

91 Some references Batchelor, G.K. (2000). An Introduction to Fluid Dynamics, Cambridge University Press. Müller, P. (1995). Ertel s potential vorticity theorem in physical oceanography. Reviews of Geophysics 33(1), Kallendorf, C., Cheviakov, A.F., Oberlack, M., and Wang, Y. (2012). Conservation Laws of Surfactant Transport Equations. Phys. Fluids 24, Kelbin, O., Cheviakov, A.F., and Oberlack, M. (2013). New conservation laws of helically symmetric, plane and rotationally symmetric viscous and inviscid flows. J. Fluid Mech. 721, Cheviakov, A.F., Oberlack, M. (2014). Generalized Ertels theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and NavierStokes equations. J. Fluid Mech. 760, A. Cheviakov (UofS, Canada) Conservation Laws II June / 35