Introduction and formulation Stability analysis Inviscid stability analysis of parallel bubbly flows Suhas S Jain ME451B Final Project Presentation, Stanford University Wednesday 12 th December, 2018 Suhas S Jain Stability of bubbly flows 1 / 23
Introduction and formulation Stability analysis Introduction and Motivation motivation objectives bubbly flows are ubiquitous in derive governing equations and nature. disturbance relations for bubble-liquid mixture. even at low void fractions, their presence can significantly change perform stability analysis of sound speed spacewise problem of inviscid attenuation characteristics bubbly shear flow, following inertia of the medium d’Agostino et al., JFM , 1997. crucial to understand the dynamical study the effect of presence of properties of the medium. bubbles. Suhas S Jain Stability of bubbly flows 2 / 23
Introduction and formulation Stability analysis Basic equations Individual phase continuity equation (IPCE): ∂ρ i α i + � ∇ · ( ρ i � u i α i ) = 0 , ∂t ρ i and α i are the density and volume fraction of phase i . Individual phase momentum equation (IPME): ∂ρ i � u i + � ∇ · ( ρ i � u i ⊗ � u i + p i 1 ) = 0 , ∂t Suhas S Jain Stability of bubbly flows 3 / 23
Introduction and formulation Stability analysis Derivation of mixture continuity equation Starting with IPCE for liquid phase: ∂ρ l α l + � ∇ · ( ρ l � u l α l ) = 0 , ∂t Rewrite, 1 Dρ l Dt + 1 Dα l Dt + � ∇ · � u l = 0 , ρ l α l If p = f ( ρ, s ), for an isentropic process, Dp Dt = c 2 Dρ Dt where, c is the speed of sound. Using this above, 1 Dp l Dt + 1 Dα l Dt + � ∇ · � u l = 0 , ρ l c 2 α l l Suhas S Jain Stability of bubbly flows 4 / 23
Introduction and formulation Stability analysis Introducing terminologies Let, β - number of bubbles per unit liquid volume n - number of bubbles per unit total volume τ - individual bubble volume α b - volume fraction of bubbles = nτ Now, # τ = 1 + gas vol. liq. vol. = tot. vol. � � 1 + βτ = 1 + liq. vol. liq. vol. # liq. vol. ∗ liq. vol # β n = tot. vol. = tot. vol = 1 + βτ βτ α b = 1 + βτ Substituting in the equation, Mixture continuity equation: � Dβτ � 1 1 Dp l Dt = � − ∇ · � u l , ρ l c 2 1 + βτ Dt l u l · � ∇ and τ = 4 / 3 πR 3 where, D/Dt = ∂/∂t + � (assuming spherical bubbles) Suhas S Jain Stability of bubbly flows 5 / 23
Introduction and formulation Stability analysis Mixture momentum equation Start with IPME for liquid phase: ∂ρ l � u l + � ∇ · ( ρ l � u l ⊗ � u l + p l 1 ) = 0 , ∂t Rewriting, � ∂ρ l α l � ∂� � u l � + � u l · � = − � � u l ∇ · ( ρ l α l � u l ) + ρ l α l ∂t + � ∇ � u l ∇ p l , ∂t Mixture momentum equation: ρ l (1 − α b ) D� u l Dt = − � ∇ p l Suhas S Jain Stability of bubbly flows 6 / 23
Introduction and formulation Stability analysis Closure Assuming volumetric mode of oscillation of the bubbles, modified Rayleigh-Plesset equation (also called as Keller-Miksis equation) �� p R ( t ) + p l ( t + R/c l ) � 1 − 1 � R + 3 R 2 � 1 − 1 � � 1 + 1 � + R ˙ R ¨ ˙ ˙ ˙ R R = R p R ( t ) ˙ c l 2 3 c l c l ρ l ρ l c l where, dots are D/Dt , p R is the liquid pressure at bubble surface and p l is the driving pressure. Boundary condition R + 4 µ ˙ p b ( t ) = p R ( t ) + 2 σ R R where, p b is the uniform bubble internal pressure, σ is the surface tension and µ is the liquid viscosity. [Keller & Miksis, J. Acoust. Soc. Am. , 1980.] Suhas S Jain Stability of bubbly flows 7 / 23
Introduction and formulation Stability analysis Final system Mixture continuity equation: � Dβτ � 1 1 Dp l Dt = � − ∇ · � u l , ρ l c 2 1 + βτ Dt l Mixture momentum equation: ρ l (1 − α b ) D� u l Dt = − � ∇ p l modified Rayleigh-Plesset equation (also called as Keller-Miksis equation): �� p R ( t ) + p l ( t + R/c l ) � 1 − 1 � R + 3 R 2 � 1 − 1 � � 1 + 1 � + R ˙ R ¨ ˙ ˙ ˙ R R = R p R ( t ) ˙ c l 2 3 c l c l ρ l ρ l c l Boundary condition: R + 4 µ ˙ p b ( t ) = p R ( t ) + 2 σ R R Suhas S Jain Stability of bubbly flows 8 / 23
Introduction and formulation Stability analysis Stability analysis of 2D parallel flows Let, u l = U ( y )ˆ e x + ˜ u ( x, y, t ) and v l = ˜ v ( x, y, t ) p l = p 0 + ˜ p ( x, y, t ) R l = R 0 + ˜ R ( x, y, t ) Let, α b → α , ρ l → ρ , c l → c , Substituting these in mass and momentum equations, linearizing and subtracting base flow and (assuming β to be uniform), disturbance mass equation � 3 α � ˆ D ˜ ˆ R 1 D ˜ p = � ∇ · � u, ˜ − ˆ ρc 2 ˆ R 0 Dt Dt where, ˆ D/ ˆ Dt = ∂/∂t + U∂/∂x disturbance momentum equation � ∂ ˜ ∂t + U ∂ ˜ u u � = − ∂ ˜ p ∂x + U ′ ˜ ρ (1 − α ) v ∂x � ∂ ˜ ∂t + U ∂ ˜ v v � = − ∂ ˜ p ρ (1 − α ) ∂x ∂y where, prime denotes ∂/∂y Suhas S Jain Stability of bubbly flows 9 / 23
Introduction and formulation Stability analysis Continued If gas is assumed to behave polytropically, then, � R 0 � 3 γ p b = p b 0 R Linearizing, � � 1 − 3 γ R 0 p b = p b 0 R Substituting these in Keller-Miksis equation and the boundary condition, linearizing and subtracting base flow, disturbance equation for bubble dynamic response + 4 µ ˙ 3 γ ˙ − 2 σ ˙ + 4 µ ¨ 3 γ ˜ − 2 σ ˜ ˜ ˜ ˜ ˜ R R R R R R 0 c = − ˜ R p ρ ¨ ˜ R + p b 0 + p b 0 R 2 R 3 R 2 cR 2 cR 0 R 0 0 0 0 0 where, dots represent ˆ D/ ˆ Dt Suhas S Jain Stability of bubbly flows 10 / 23
Introduction and formulation Stability analysis Normal mode assumption Now, making an ansatz for the disturbance, u ( y ) e i ( kx − ωt ) ˜ u = ˆ v ( y ) e i ( kx − ωt ) ˜ v = ˆ p ( y ) e i ( kx − ωt ) p = ˆ ˜ R = ˆ ˜ R ( y ) e i ( kx − ωt ) Substituting these in the disturbance equations, disturbance mass equation v ′ = − i 3 γ R + i ω L ω L ˆ ik ˆ u + ˆ ρc 2 ˆ p R 0 disturbance momentum equation u + U ′ ˆ ρ (1 − α )( − iω L ˆ v ) = − ik ˆ p p ′ ρ (1 − α )( iω L ˆ v ) = ˆ where, ω L = ω − Uk is the Lagrangian frequency. Suhas S Jain Stability of bubbly flows 11 / 23
Introduction and formulation Stability analysis Continued disturbance equation for bubble dynamic response � � � 1 + i ω L R 0 � ˆ p ω 2 ω 2 ˆ − − iω L λ + R = − L b c ρR 0 � �� � ���� ���� damping inertial compressiblity where, ω 2 L R 0 4 µ � � λ = + thermal = 0 − ρR 2 c 0 � �� � � �� � acoustical viscous is the damping coefficient and b = p b 0 3 γ − 2 σ ω 2 ρR 2 R 3 0 0 is the natural frequency of the bubble Suhas S Jain Stability of bubbly flows 12 / 23
Introduction and formulation Stability analysis Dispersion relation for homogeneous medium Let both x and y be homogeneous. Repeating the whole process again, v and ˆ Then eliminating ˆ u , ˆ R from the 4 disturbance equations ⇒ wave equation for ˆ p , �� 3 α �� (1 − α )(1 + iω R 0 c ) � � b ) + (1 − α ) ω 2 − k 2 x + k 2 p = 0 ˆ ( − ω 2 − iωλ + ω 2 y R 2 c 2 0 � �� � =0 ⇒ dispersion relation speed of propagation of harmonic disturbance ω in the bubbly mixture medium (1 − α )(1 + iω R 0 c ) m ( ω ) = 3 α 1 b ) + (1 − α ) R 2 ( − ω 2 − iωλ + ω 2 c 2 c 2 0 Suhas S Jain Stability of bubbly flows 13 / 23
Introduction and formulation Stability analysis Going back to the parallel flow setup Eliminating ˆ R and ˆ p from the 4 disturbance equations, equivalent Rayleigh system for bubbly flows v − i U ′′ U ′ u ′ = ik ˆ � � u − U ′ ˆ ˆ v − i ˆ iω L ˆ v ω L kc 2 m ( ω L ) ω L v ′ = − ik ˆ � � u − U ′ ˆ ˆ u + iω L ˆ v kc 2 m ( ω L ) In the limit of c m → ∞ , v − i U ′′ u ′ = ik ˆ ˆ ˆ v ω L v ′ = − ik ˆ ˆ u Eliminating ˆ u , it reduces to the classical Rayleigh equation, ( U − c )( D 2 − k 2 )ˆ v − U ′′ ˆ v = 0 where, D is ∂/∂y Suhas S Jain Stability of bubbly flows 14 / 23
Introduction and formulation Stability analysis Base flow and boundary conditions Base flow: Inviscid shear layer U ( y ) = U 1 + U 2 + U 2 − U 1 tanh ( y δ ) 2 2 Boundary conditions: When y → ±∞ ⇒ U = const , then the system reduces to, u ′ = ik ˆ ˆ v ω 2 v ′ = − ik ˆ L ˆ u + i m ( ω L ) ˆ u kc 2 This admits a close form solution as, Asymptotic solutions v = Ae ± y ( k 2 − ω 2 L /c 2 m ) 1 / 2 ˆ ik m ) 1 / 2 e ± y ( k 2 − ω 2 L /c 2 m ) 1 / 2 u = ± A ˆ ( k 2 − ω 2 L /c 2 where A is an arbitrary complex constant. Suhas S Jain Stability of bubbly flows 15 / 23
Introduction and formulation Stability analysis Solution procedure Method: Shooting method. Spacewise problem: complex k and real ω is assumed. Procedure Guess a complex eigenvalue k . Choose A such that initial conditions at y = − nδ simplifies, where n ≫ 1 ( n = 5 in this project) v = 1 ˆ ik ˆ u = ( k 2 − ω 2 L /c 2 m ) 1 / 2 Integrate upto y = nδ (using RK4 in this project). Check if the solution is continuous with the asymptotic solution at y = nδ ik u = − ˆ m ) 1 / 2 ˆ v ( k 2 − ω 2 L /c 2 Iteratively correct eigenvalue k until convergence (using 2D Newton-Raphson method in this project) Suhas S Jain Stability of bubbly flows 16 / 23
Recommend
More recommend
