TsAGI works Jour., 2294, Moscow 1985 (in Russia) 10. Y.F. Zhuravlev, V.M. Lapin The structure of high-pressure gas jets inside cocurrent water flow. -- TsAGI' works Jour., 2685, Moscow 2009 (in Russia) SOME PROBLEMS OF SUPERCAVITATIONAL FLOW CFD MODELING Yury F. Zhuravlev Anton V. Varyukhin Central Aero-hydrodynamic Institute (TsAGI) Radio str., 17, 105005 Moscow, Russia World Maritime Technology Conference, WMTC2012, May 29-June 1, 2012, Saint-Petersburg, Russia Introduction Supercavitating body motion underwater is accompanied by gas jets flowing inside a cavity. It is connected with blowing of gas to hold required cavity size The gas jets lead to a nonuniform pressure distribution along the cavity boundary. As an example inclined jets lead to non axisymmetric disturbances of the cavity flow Supercavity modeling numerical methods 1. Method based on the numerical integration of the multiphase Navier-Stocks' system of equations 2. Supercavity flow modeling based on the Laplace's problem solution for the external flow and the numerical integration of Navier-Stocks equations for the internal flow (gas flow inside the cavity) 1. Method based on the numerical integration of the multiphase Navier-Stocks system of equations 1 ( ) ( ) p N r r U t α α α α α α αβ β ρρ = ∂ + ∇ = Γ ∂ ∑ Continuity equation: Momentum equations: Energy equation: ( ) ( ) ( ) ( ) 0 0 0 0 1 2 3 p N T p r h r r U h r T r U U U U h h Q t t α α α α α α α α α α α α α α α α α α α αβ β β α β α β β ρ ρ λ μ δ = ∂ ∂ − + ∇ − ∇ − ∇ ∇ + ∇ − ∇ = Γ − Γ + ∂ ∂ ∑ 1 ( ) ( ( )) ( ( ( ) )) ( ) p N T r U r U U r p r U U U U M t α α α α α α α α α α α α α α β α β αβ β α β ρ ρ μ + + = ∂ + ∇ × = ∇ + ∇ ∇ + ∇ + Γ − Γ + ∂ ∑ 1 1 p N r α β = = ∑ Constrain of volume fractions , 1... p p p N α α = = Equality of pressures Homogeneous approach 1 p N r α α α ρ ρ = = ∑ 1 p N r α α α μ μ = = ∑ , 1... p U U N α α = = Inhomogeneous approach • Inhomogeneous multiphase approach requires three times more equation to be solved than homogeneous approach Mass flowrate from α to β Volumetric force on from others α Heat flux to from others α Source terms ( ) D M C A U U U U α αβ αβ αβ β α β α βα ρ ≠ = − − ∑ ( ) A Q T T Nu d αβ αβ α β α αβ βα λ αβ λ ≠ = − ∑ rr αβ α α β β ρ ρ ρ = + rr αβ α α β β λ λ λ = + rr A d αβ αβ αβ = Re 24 1 0,44 Re 6 D C αβ αβ αβ = + + 1 2 Re 2 Nu αβ αβ = + Re U U d αβ α β αβ αβ αβ ρ μ − = 2 [0, ] (1/2 ) lv v v v lv l C r Min p p U t ρ ρ ∞ ∞ − Γ = 2 (1 ) vl v v v vl C r r t ρ ∞ − Γ = is contact area between phases per unit volume is mean thermal conductivity is mean density 1. Method based on the numerical integration of the multiphase Navier-Stocks system of equations Vapour cavity modeling Temperature of vapour and water is Т=288 К Hydrostatic pressure is Pa 5 10 h p = Water velocity is 70m/s Cavitator diameter is mm 0 V = 10 n d = Homogeneous approach Inhomogeneous approach Water volume fraction distribution Distribution of cavitation number Water volume fraction distribution ( ) q p p h − = σ Distribution of cavitation number ( ) q p p h − = σ • In case of homogeneous approach cavity length is shorter and Efro's re-entrained jet is absent Efro's re-entrained jet Ventilated cavity modeling. Inhomogeneous approach Temperature of vapour, water and gas is Т=288 К Hydrostatic pressure is Па 5 10 h p = Water velocity is 30m/s Cavitator diameter is mm 0 V = 10 n d = Non-dimensional volume flowrate 2 0 3 n Q Q Vd = = σ / n x x R = σ Time, [s] Cavitation number distribution along cavity • Flow is unsteady • Efro's re-entrained is good modeled •
Cavitational number is almost constant along cavity Ventilated cavity modeling. Comparison homogeneous and inhomogeneous approaches Homogeneous approach Inhomogeneous approach Water volume fraction distribution Water volume fraction distribution Temperature of vapour, water and gas is Т=288 К Hydrostatic pressure is Па 5 10 h p = Water velocity is 30m/s Cavitator diameter is mm 0 V = 10 n d = Non-dimensional volume flowrates 2 0 3 n Q Q Vd = = Distribution of cavitation number ( ) q p p h − = σ Distribution of cavitation number ( ) q p p h − = σ • In case of homogeneous approach the cavity shape is not symmetry Comparison CFD results with experimental data Vapour cavity modeling Ventilated cavity modeling 0 1000 TK = 0 500 TK = Ventilated cavity modeling. Hot gas flow modeling • Pumped gas inside cavity is cooled rather intensively • CFD results approve the earlier hypothesis that gas flow inside cavity can be considered isothermal Cavity boundary Gas is pumped behind the cavitator Ventilated cavity modeling. Cavity bow profiles 0.005 n R = Cavitator radius 1/3 3 ( ) 1 , 3 nn n x R x R x R R = + Logvinovich's law for cavity profile near flat cavitator • All approaches give the results that are in good correspondence with experimental data Cavity at body's entry into water 0 2 4 6 8 10 12 14 16 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 CFD Experiment R h h 2 = b V F F 2 = B.C. Inlet (Velocity, volume fraction) B.C. Outlet - Water flow velocity V 0 =0, 5, 7m/s - Depth h=0.5m - Diameter of nozzle critical section d cr =3.0 mm - Diameter of nozzle outlet section d out =3.7mm - Pressure behind nozzle P 0 =10 atm High pressure bottle Nozzle Pipe for measuring pressure, water velocity and water volume fraction along jet axis Water free surface Water volume fraction and velocity measuring instrument s m V / 5 0 = Shutter is 0.001sec Gas jets submerged in the co-current water flow Experiment by Zhuravlev, Lapin TsAGI 1984 Computational domain Гр.усл. Вход Объёмная доля жидкости r w =1 Скорость V 0 Гр.усл. Выход BC. Inlet Объёмная доля воздуха r a =1 Массовый расход G Mach number distribution Water volume fraction distribution CFD modeling of gas jets Jet geometry 0 2 4 6 8 10 12 14 16 18 0 5 10 15 20 25 30 35 40 Exp. V0 = 0 m/s Exp. V0 = 5 m/s Exp. V0 = 7 m/s Num. V0 = 0 m/s Num. V0 = 5 m/s Num. V0 = 7 m/s 0 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 15 20 25 30 35 40 Exp. V0 = 0 m/s Exp. V0 = 5 m/s Exp. V0 = 7 m/s Num. V0 = 0 m/s Num. V0 = 5 m/s Num. V0 = 7 m/s out d x x = wat out d d d = Water volume fraction along jet axis out d d d = Comparison CFD results with experimental data • Calculated water volume fraction and velocity distributions are in good agreement with experimental data 2. Method based on the Laplace's problem solution for the external flow and the numerical integration of Navier-Stocks equations for the internal flow (gas flow inside the cavity) - Water is non-viscose and incompressible - Gas and liquid are not get mixed, gases do not dissolve in water - The liquid evaporation can be neglected, because the saturated gas pressure is much smaller than the cavity pressure - The liquid velocity does not change on the cavity surface due to viscous interaction with the gas jet -
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