D REW , 1983) Following Drew (1983) ( k k ) + ( k k v k ) = 0 - PowerPoint PPT Presentation

P ARTICLE - LADEN FLOWS : SOME CONUNDRUMS Peter Duck Head, School of Mathematics, University of Manchester ICASEr 1987 - 1994 Collaborators: Rich Hewitt and Mike Foster

S OME HISTORY Numerous papers (of variable quality) have been published Annual Review articles by Marple (1970) and Drew (1983) Einstein (1906) investigated these flows An early investigation involving the stability of plane Poiseuille flow made by Saffman (1962) Michael (1968) investigated the (inviscid) dusty flow past a sphere Hydrodynamics of Suspensions Ungarish (1993) The Dynamics of Fluidized Particles Jackson (2000)

(C OMPREHENSIVE ) EQUATIONS OF MOTION ( A LA D REW , 1983) Following Drew (1983) ∂ ( α k ρ k ) + ∇ · ( α k ρ k v k ) = 0 ∂ t ∂ ( α k ρ k v k ) � � + ∇ · ( α k ρ k v k v k ) − α k ∇ p k + ∇ · = α k ( τ k + σ k ) ∂ t ( p k , i − p k ) ∇ α k + M k + k = 1: particles, k = 2: fluid. τ k is (approximately) stress tensor, σ k turbulent stress tensor, p k , i pressure at interface, M k ‘interfacial’ force density’; α , ρ , v volume fraction, density and velocity fields.

S IMPLIFICATIONS Turbulent stresses σ k = 0 Drew (1983) states p k , i = p k for non-acoustic problems Drew (1983) states p 1 = p 2 + p c , where p c pressure due to collisions; assume p 1 = p 2 , and constant. Assume τ 1 = 0 for solid particles Must have α 1 + α 2 = 1, M 1 + M 2 = 0 Will consider 3 problems

N ON - DIMENSIONAL EQUATIONS OF MOTION u f , u p fluid, particle velocities − ∂α ( 1 − α ) u f � � ∂ t + ∇ · = 0 ∂ u f ∂ t +( u f ·∇ ) u f = −∇ p + 1 1 + βα � � ( 1 − α ) e 1 − α ( u p − u f ) , 1 − α ∇· Re ∂α ∂ t + ∇ · ( α u p ) = 0 , ∂ u p ∂ t + ( u p · ∇ ) u p = − 1 γ ∇ p + β γ ( u f − u p ) Here Re = UL /ν , β = ( 9 ν L ) / ( 2 Ud 2 ) (Stokes drag), γ = ρ p /ρ f , e rate of strain tensor for fluid.

P ROBLEM 1: STEADY DUSTY INVISCID FLOW OVER A CIRCULAR CYLINDER Cylinder version of sphere problem considered by Michael (1968) - fluid affects particles but not v.v. Polar coordinates ( r , θ ) , velocity ( u , v ) , as α → 0: � T � ( 1 − 1 r 2 ) cos θ, − ( 1 + 1 ( u f ( r , θ ) , v f ( r , θ )) T = r 2 ) sin θ Then γ → ∞ (heavy particles), γ/β = O ( 1 ) : ∂θ − v 2 ∂ u p ∂ r + v p ∂ u p = β p u p γ ( u f − u p ) , r r ∂ v p ∂ r + v p ∂ v p ∂θ + u p v p = β u p γ ( v f − v p ) , r r 1 ∂ r ( r α u p ) + 1 ∂ ∂ ∂θ ( α v p ) = 0 r r

P ARTICLE PATHS , β/γ = 5 5 4 3 2 1 0 -4 -2 0 2 4

S EPARATION ANGLE 3.2 3 2.8 2.6 2.4 θ sep 2.2 2 1.8 1.6 1.4 0 1 2 3 4 5 6 7 8 β/γ ‘Separation’ angle θ sep for increasing values of inter-phase drag parameter β/γ .. Note as β/γ → 0 + , θ sep → π/ 2 and as β/γ → 8 − , θ sep → π , as shown as dashed lines

C ONDITIONS ALONG θ = π 1 2 0.9 0 0.8 -2 0.7 0.6 -4 0.5 u p Inc. β/γ log ( α ) -6 0.4 -8 0.3 0.2 -10 Inc. β/γ 0.1 -12 0 -0.1 -14 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 x x As r → 1, can show u p = u p 0 + ( r − 1 ) u p 1 + . . . , where � � � u p 1 = β 1 − 8 γ u p 0 = 0 , for 1 + β/γ < 8 , 2 γ β u p 0 � = 0 , u p 1 = − β/γ for β/γ > 8

I SSUES ARISING Particles can ‘penetrate’ cylinder surface Solution discontinuous at β/γ = 8 ‘Shadow’ regions Since on θ = π , r α u p = constant, if u p → 0, α → ∞ : violates α << 1 condition A mixed elliptic/hyperbolic system

P ROBLEM 2: SETTLING UNDER GRAVITY Consider the stationary (1D) distribution of heavy ( γ >> 1) dust phase ‘settling’ under uniform gravity in an upwards propagating fluid; the dust-phase weight balanced by upwards motion of fluid. Fluid particle free and moving with constant speed V 0 for y < 0, y = 0 is location of stationary front. α ( y ) V 0

I NCLUDE ( FLUID ) VISCOSITY , AND ( NOTIONALLY ) ASSUME α = O ( 1 )) . Problems of this type considered by Druzhinin (1994, 1995), with some inconsistencies. Particles affect fluid and v.v. Problem reduces to V p = 0 (( 1 − α ) V f ) ′ = 0 α ′ V ′ V f 1 − α = γ − 1 + 2 f − 2 f V f V ′ ReV ′′ f + Re 1 − α Assume V f ( y = 0 ) = V 0 (constant) and α ( y = 0 ) = 0.

(B ASEFLOW ) RESULTS FOR V ( y ) AND α ( y ) 0.8 a 0.7 V f ( y ) 0.6 0.5 0.4 0.3 α ( y ) 0.2 0.1 y 0 0 1 2 3 4 5 7 V f ( y ) b 6 5 4 3 2 α ( y ) 1 y 0 0 1 2 3 4 5 V f ( y ) 3 c 2.5 2 1.5 α ( y ) 1 0.5 y 0 0 1 2 3 4 5 (a): γ = 2, V 0 = 0 . 5, Re = 1 , 2 , 4 , 8 , 16 , 32 (solid lines), the dashed lines show the Re ≫ 1 solution, (b): V 0 = 1 . 5, Re = 20, γ = 4 , 8 , 16 , 32 and (c): γ = 4, Re = 20, V 0 = 0 . 5 , 1 , 1 . 5 , 2 , 2 . 5

S PATIAL LINEAR STABILITY IN THE INVISCID LIMIT Can perturb the inviscid system for a steady base flow via α ( y ) e − i ω t , α = α B ( y ) + ǫ ˜ v f ( y ) e − i ω t , V f = V fB ( y ) + ǫ ˜ v p ( y ) e − i ω t , V p = 0 + ǫ ˜ Here ǫ << 1, and ω is a real frequency. Focus on y → ∞ : � V 0 α B → α ∞ = 1 − γ − 1 1 V f → V ∞ = (( γ − 1 ) V 0 ) 2

L INEAR STABILITY IN THE INVISCID LIMIT ( CONTINUED ) v p ) e iky Then α ( y ) , ˜ v f ( y ) , ˜ v p ( y )) = (ˆ v f , ˆ α, ˆ Further supposing (˜ � � ω 2 ( γ + α ∞ ( 1 − γ )) � − 2 ω 2 � ω k 2 + k + + − = 0 V ∞ iV ∞ ( 1 − α ∞ ) α ∞ V 2 iV 2 ∞ α ∞ ( 1 − α ∞ ) ∞ Considering the limit ω → ∞ , we write k = ω K , K = O ( 1 ) , i � 1 γ ( 1 − α ) K = ± V ∞ v ∞ α ∞ Spatial growth is  − 1  1  � 1  2 V 0 � 2 ωγ 2   y exp  1 −     1  3 ( γ − 1 )    4 V  4 ( γ − 1 ) 0 Spatial growth rate proportional to frequency - implies problem ill-posed.

L INEAR STABILITY , FINITE Re Equation for k now a cubic: 2 α ∞ V ∞ iV 2 � � � � ∞ α ∞ 2 α ∞ k 3 + k 2 ω ( 1 − α ∞ ) − ( 1 − α ∞ ) ω Re Re ( 1 − α ∞ ) ( 1 − α ∞ ) 2 ω − 2 iV ∞ α ∞ 2 α ∞ V ∞ � � � � � � α ∞ 1 + k i ω + γ + = 0 − ( 1 − α ∞ ) 2 1 − α ∞ 1 − α ∞ As ω → ∞ , Re = O ( 1 ) , two families:- � 1 � − ( i ωγ ( 1 − α ∞ ) + α ∞ ) Re 2 k → ± 2 α ∞ − i γ V ∞ ( 1 − α ∞ ) Re ω k → V ∞ 2 α ∞

P ROBLEM 3: B OUNDARY LAYERS IN A DILUTE PARTICLE SUSPENSION Foster, Duck & Hewitt (2006) - mixed parabolic/hyperbolic problem g u e ∼ x m | 2 θ | = 2 m π m + 1 Flow geometry appropriate for Falkner–Skan-type edge conditions, although solutions not restricted to have self similarity. Assume that local gravitational forcing is aligned as shown and thus the upper boundary layer is such that K > 0 whilst the lower boundary layer has K < 0, where K = gLRe 1 / 2 ( 1 − 1 γ ) U 2 ∞

D USTY BOUNDARY - LAYER EQUATIONS Usual boundary-layer scalings uu x + vu y + ¯ p x u yy − βα ( u − u p ) , = β u p u px + v p u py γ ( u − u p ) , = β u p v px + v p v py γ ( v − v p ) − K cos θ, = u x + v y = 0 , u p α x + v p α y − α ( u px + v py ) . = u = v = 0 on y = 0 , u → u e ( x ) as y → ∞ Choice of boundary conditions for the particle phase is somewhat subtle; notionally u p → u pe ( x ) , and α → α e ( x ) , for y → ∞ . Will consider U e ( x ) = x m , 0 ≤ m ≤ 1

T HE OUTER FLOW AND CONDITIONS AT THE BOUNDARY - LAYER EDGE pe = β e + β γ E e = − β u pe u ′ γ ( u e − u pe ) , u pe E ′ e + E 2 γ u ′ u pe α ′ e , e + α e D e = 0 , (1) where E e ( x ) = ∂ v p /∂ y ( y → ∞ ) , u pe ( x ) = u p ( y → ∞ ) and D e ( x ) = D ( y → ∞ ) = u ′ pe + E e are the relevant functions evaluated as y → ∞ Also e + β pe ) 2 + E 2 u pe D ′ γ D e + ( u ′ e = 0 . pe + β = β � � u pe u ′ γ u e , γ d α ′ � dx + β � � � e u pe u pe = ( u ′ pe ) 2 + E 2 e . γ α e For given fluid edge behaviour u e ( x ) , can determine streamwise particle motion u pe ( x ) , then particle motion normal to boundary E e , and finally external volume fraction α e ( x ) .

E DGE RESULTS 14 1.7 α 0 ) 12 1.6 log ( α e 1.5 10 1.4 u pe 8 1.3 1.2 6 1.1 4 1 0.9 (a) (b) 2 0.8 0.7 x 0 x 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 Development of the edge quantities (a) u pe and (b) α e ; solid m = . 50; dashed m = . 211; dotted m = . 10 , all with β/γ = 1; K not relevant here. x − x 0 if m > m crit - violates α << 1 1 Can show α e ∼

B OUNDARY - LAYER RESULTS (a): K = 0 (b): K = 10 u p ( η = 0 ) 2 1 U η ( η = 0 ) 0.5 1.5 βα ( η = 0 ) u p ( η = 0 ) 0 1 v p ( η = 0 ) / K U η ( η = 0 ) -0.5 0.5 βα ( η = 0 ) -1 x x 0 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 Development of wall values with x for m = 0, β/γ = 1. In case (b), leading-order asymptotic forms for x → ∞ are shown for x > 4 . 5;

S INGULARITIES INSIDE THE BOUNDARY LAYER Taking K = 0 (for simplicity - particle flow can be solved explicitly on y = 0): α w = α 0 α 0 U pw > 0 . = , U pw 1 − β x γ Therefore volume fraction is singular part way along wall, and so model breaks down (Wang & Glass, 1988 continued their computation through the singularity). Can be a ‘race’ between inner and outer singularities.