Practical Ship Hydrodynamics

Numerical example for BEM 257
At the free surface ⊲z D total ⊳ the pressure is constant, namely atmospheric
pressure ⊲p D p0⊳:
D⊲p p0⊳
Dt
D
∂⊲p p0⊳
∂t
C ⊲r total r⊳⊲p p0⊳ D 0
Bernoulli’s equation gives at the free surface ⊲z D total ⊳ the dynamic boundary
condition:
total
t
1
C
2 ⊲r total ⊳ 2 C g total C p D 1
2 V2 C p0
The kinematic boundary condition gives at z D total :
D total
Dt
D ∂
∂t
total C ⊲r total r⊳ total D total
z
Combining the above three equations yields at z D total :
total
tt
C 2r total r total
t
Cr total r⊲ 1
2r total ⊳ 2 ⊳ C g total
z D 0
Formulating this condition in ⊲0⊳ and ⊲1⊳ and linearizing with regard to
instationary terms gives at z D total :
⊲1⊳
tt C 2r ⊲0⊳ r ⊲1⊳
t Cr ⊲0⊳ r⊲ 1
2⊲r ⊲0⊳ ⊳ 2 Cr ⊲1⊳ r ⊲0⊳ ⊳
Cr ⊲1⊳ r⊲ 1
2⊲r ⊲0⊳ ⊳ 2 ⊳ C g ⊲0⊳ ⊲1⊳
z C g z D 0
We develop this equation in a linearized Taylor expansion around ⊲0⊳ using the
abbreviations Ea, Ea g ,andBfor steady flow contributions. This yields at z D ⊲0⊳ :
⊲1⊳
tt C 2r ⊲0⊳ r ⊲1⊳
t Cr ⊲0⊳ Ea g Cr ⊲0⊳ ⊲r ⊲0⊳ r⊳r ⊲1⊳
Cr ⊲1⊳ ⊲Ea CEa g ⊳ C Ba g
3 ⊲1⊳ D 0
The steady boundary condition can be subtracted, yielding:
⊲1⊳
tt C 2r ⊲0⊳ r ⊲1⊳
t Cr ⊲0⊳ ⊲r ⊲0⊳ r⊳r ⊲1⊳ Cr ⊲1⊳ ⊲Ea CEa g ⊳ C Ba g
3 ⊲1⊳ D 0
⊲1⊳ will now be substituted by an expression depending solely on ⊲0⊳ , ⊲0⊳ ⊲ ⊲0⊳ ⊳
and ⊲1⊳ ⊲ ⊲0⊳ ⊳. To this end, Bernoulli’s equation is also developed in a Taylor
expansion. Bernoulli’s equation yields at z D ⊲0⊳ C ⊲1⊳ :
total
t
1 C 2⊲r total ⊳ 2 C g total D 1
2V2 A truncated Taylor expansion gives at z D ⊲0⊳ :
⊲1⊳
t
C 1
2 ⊲r total ⊳ 2 C g
⊲0⊳ 1
2V2 C ⊲r total r total
z C g⊳ ⊲1⊳ D 0
Formulating this condition in ⊲0⊳ and ⊲1⊳ , linearizing with regard to instationary
terms and subtracting the steady boundary condition yields:
⊲1⊳
t Cr ⊲0⊳ r ⊲1⊳ C a g
3 ⊲1⊳ D 0