Modeling of the Unsteady Separated Flow over Bilge Keels of FPSO Hulls under Heave or Roll Motions. Yi-Hsiang Yu 09/23/04 Copies of movies/papers and today’s presentations may be downloaded from Http://cavity.ce.utexas.edu/kinnas/fpso/. Introduction.
Copies of movies/papers and today’s presentations may be downloaded from
excessive roll motions, and the installation of bilge keels has been widely used as an effective and economic way of mitigating the roll motions of hulls.
Non-Dimensional Governing Equation (Navier-Stokes Equation & Continuity Equation)
where U represents the velocity; Q is the force term; and R indicates the viscous term. The definitions of the column matrices for the Navier-Stokes equation are given as
where the Reynolds number is define as Re = Umh/ν; and the length scale, h, is a representative length in the problem being solved.
(Collocated variable, non-staggered grid arrangement)
According to the integral formulation of the Navier-Stokes equation and to the Gauss divergence theorem, a semi-discrete integral formulation of the momentum equation can be given as
where Sijis the area of the cell; and ds represents the length of each cell face. A cell center based scheme is applied where “i,j” is the center of the cell (non-staggered grid: unknown value u, v, p are located at the cell center).
when calculating the flux, the value on the cell face (at D) is needed. It can be obtained from Taylor series expansion.
where f represents the summation of the convective terms, the viscous terms and the pressure terms at the present time step n and the next time step n+1.
where p’ is the pressure correction, V’faceis the velocity correction term, әp’ /әnis the pressure correction derivative with respect to the normal direction of the cell face, V*face= (u*; v*) is the predicted velocity vector obtained from the momentum equation.
Since our 2-D Navier-Stokes solver uses non-staggered grid, the scheme has to be modified somewhat to avoid the checkerboard oscillation problem.
where aijis the coefficient of the unknown velocity in the momentum equation; "av" indicates the average value obtained from the cell center value; and "d" represents the value calculated directly at the face center.
The momentum equation can be rearranged as
where u* is the unknown velocity at T = n + 1; the coefficient “a” is also a function of the velocity at T = n+1 which can be obtained from the previous iteration; and dijis the coefficient of pressure in the momentum equation.
When the grid is moving,
additional terms need to
be taken into account.
where (ugrid, vgrid) is the velocity of the moving grid; and
represents the total change in the value of u with both increment in time and the corresponding change in the location of the point. When the above equation is substituted into the momentum equation
or without the bilge keels.
bilge keels and the effect of
the free surface
Kc=UmT/h (0.5 < Kc < 5)
FVM Motion at t/T=0.25
The pressure distribution along the submerged hull without bilge keels