Method for analyzing friction reduction effects of bubbles in friction-reducing ship

A method for analyzing effects of generating bubble jets in a flow field near a ship surface on reducing skin-friction in a cruising ship comprising the steps of:

• performing a Fourier transform on kinetic equations (1) and (2) for a bubble moving in x- and y-directions, representing a fluid flow direction and a direction at right angles to said ship surface, respectively, in terms of an added mass of a bubble mA, a bubble diameter db, a dynamic liquid viscosity coefficient vL, and obtaining an expression for a gain in x-direction Gx and an expression for a gain in y-direction Gy comprised of a bubble time-constant T and a turbulence frequency ωL, based on equations (3) and (4) respectively;
• obtaining from equation (16) a resistive force ΔRv acting on a bubble in a high frequency band region where a product of a bubble time-constant T and a turbulence frequency ωL is greater than 1, in terms of a normal wall constant κ in the wall law in non-bubbly flow, a fluid density ρL, a dynamic fluid viscosity coefficient vL, and a time-averaged velocity in x-direction uL, by assuming that a turbulence cycle 2π/ωL in said flow field is equal to said integral time-scale T.L;
• obtaining a shear stress decrement τt caused by said resistive force ΔRv from equation (17);
• obtaining a mixing length decrement lmb from equation (27) in terms of an empirical constant a, said bubble diameter db, said dynamic viscosity coefficient vL, a frictional velocity Ut and a near-wall local void fraction αW, by comparing said equation (17) with equation (22) which expresses a shear stress decrement r, that is assumed to be produced by a mixing length decrement lmb;
• obtaining a revising wall constant κ2 in the wall law as in equation (33) by using said equation (27), and obtaining an expression as in equation (34) for an operative wall constants κ1 for bubbly flow, in terms of said empirical constant a and said bubble diameter db as parameters, by subtracting said revised wall constants κ2 from said normal wall constant κ;
• deriving equation (44) relating said normal wall constant κ to a local friction factor Cf0 in non-bubbly flow, and equation (45) relating said operative wall constant κ1 to a local friction factor Cf in bubbly flow, according to equation (36) for non-bubbly flow velocity and equation (40) for bubbly flow velocity, assuming that a fluid velocity distribution obeys a logarithmic law and that a position parameter y is represented by a turbulent boundary layer thickness δ;
• subtracting said equation (44) from said equation (45) to obtain equation (48), and expanding a series expression (Cf/Cf0)½ in equation (48) about 1 so as to derive equation (49) comprised of first-order expansion terms while also expanding a bottom-layer term vL/Ut in said equation (48) for an operative wall constants κ1 for bubbly flow, thereby deriving equation (50);
• and
• obtaining an analytical solution (55) for a skin-friction ratio Cf/Cf0 in said high frequency band region by substituting said equation (50) in said equation (33) to yield a first result and substituting said first result into said equation (34) to yield a second result and substituting said second result into said equation (49) and solving to obtain an expression for said analytical solution;
• wherein equations referenced above are listed as follows;
• m A 3 ⁢ πμ L ⁢ d b ⁢ X ¨ + X . = u L ′ ( 1 ) m A 3 ⁢ πμ L ⁢ d b ⁢ Y ¨ + Y . = v L ′ ( 2 ) X . ^ u ^ L ′ = 1 1 + T 2 ⁢ ω L 2 = G X ( 3 ) Y ^ v ^ L ′ = 1 1 + T 2 ⁢ ω L 2 ⁢ 1 ω L = G Y ( 4 ) Δ ⁢ R ^ v = 27 4 ⁢ π ⁢ κρ L ⁡ ( ϵ L ⁢ T * L ) 2 ⁢ ( ∂ u _ L ∂ y ) 2 ( 16 ) τ t = 81 π 2 ⁢ κρ L ⁡ ( v L ⁢ T * L d b ) 2 ⁢ α w ⁡ ( ∂ u _ L ∂ y ) 2 ( 17 ) τ t = 2 ⁢ ρ L ⁢ l m0 ⁢ l mb ⁡ ( ∂ u _ L ∂ y ) 2 ( 22 ) l mb = ( v L / U τ d b / a ) 2 ⁢ α w ⁢ y ( 27 ) κ 2 = ( v L / U τ d b / a ) 2 ⁢ α w ( 33 ) κ 1 = ⁢ κ - κ 2 = ⁢ κ - ( v L / U τ d b / a ) 2 ⁢ α w ( 34 ) u 0 + = 1 κ ⁢ log ⁢ ⁢ y 0 + + B ⁢ ⁢ where ⁢ ⁢ B ⁢ ⁢ is ⁢ ⁢ a ⁢ ⁢ constant ( 36 ) u + = 1 κ 1 ⁢ log ⁢ ⁢ y + + B ( 40 ) 2 C f0 = 1 κ ⁢ log ⁢ ⁢ δ 0 + + B ( 44 ) 2 C f = 1 κ 1 ⁢ log ⁢ C f C f0 ⁢ δ 0 + + B ( 45 ) 2 C f0 ⁢ C f0 C f - 2 C f0 = - 1 κ 1 ⁢ log ⁢ C f0 C f + ( 1 κ 1 - 1 κ ) ⁢ log ⁢ ⁢ δ 0 + ( 48 ) 2 C f0 ⁢ C f0 C f - 2 C f0 = - 1 κ 1 ⁢ ( 1 - C f0 C f ) + ( 1 κ 1 - 1 κ ) ⁢ log ⁢ ⁢ δ 0 + ( 49 ) v L U τ = v L U ⁢ 2 C f = C f0 C f ⁢ v L U ⁢ 2 C f0 ( 50 ) C f C f0 = { 1 + 2 / C f0 ⁢ κ 10 - 2 ⁢ ( log ⁢ ⁢ δ 0 + ) ⁢ κ 20 / κ 1 + 2 / C f0 ⁢ κ 10 - ( log ⁢ ⁢ δ 0 + ) ⁢ κ 20 / κ } ⁢ ( 1 - α w ) . ( 55 )