2.6 Solutions for particular two dimensional problems
2.6.5 Arbitrary motion in time domain: simple source method
b = Imj ( −F˜ ω ) (2.62)
It is usual to talk of added mass and damping for a particular direction of force and mode of motion. Thus it is implied in equations (2.61) and (2.62) that the heave or sway force is indicated by the appropriate choice of ˜F in equation (2.60), with ˜φbeing the solution for heave or sway motion as determined by the appropriate choice of boundary condition represented by equation (2.55), giving a total of eight possible coefficients represented by equations (2.61) and (2.62).
2.6.5
Arbitrary motion in time domain: simple source method
The implementation of this method was restricted to fully submerged bodies. This was for two reasons: the nature of arbitrary motion excludes linearisation of the free surface based on the small motion assumption unless one wishes to restrict the possible motions to small motions about a mean position, and surface piercing bodies involve additional geometric manipulations and programming difficulties that were not justified for the applications envisaged.
The source function in this problem, as in the steady motion problem, equation (2.37), is f(z) = 2Qπ(ln (z−c) + ln (z−c)), with the second logarithm (image term) omitted for free¯ surface sources.
Body equations
As for all linear simple source methods there are two sets of equations, those on the body and those on the free surface, giving rise to a matrix equation identical to (2.36) in the steady motion problem. Furthermore the body boundary condition, being a kinematic condition, is identical to that for steady motion with the only exception being that the body velocity is now a function of time15. Sub-matrices [ABB], [ABF] and {RB} may therefore be evaluated using equations
(2.38), (2.39) and (2.40).
Free surface equations
The linearised free surface boundary condition to be satisfied is ∂∂t2φ2 +g∂φ∂y = 0 (equation (2.12)), and, as in the steady motion problem, the second term in this equation is very easily evaluated using ∂φ∂y = −2Q.
There are two approaches to evaluating ∂∂t2φ2: analytical or numerical. The former would at first seem preferable, but it gives rise to terms involving the derivative of the source strength,
∂2Q
∂t2, which, due to the time-discretisation and the fact that Q(t) is not previously known, must be numerically evaluated anyway. Neither method has a clear advantage over the other therefore, and the numeric approach will be taken here while the analytic one will be used in the equivalent Green function solution in the next chapter.
The numerical approximation to ∂∂t2φ2 must contain values of φfrom past and current time steps, with the former being known explicitly and the latter being in terms of the unknown source strengths at the current time step. The three point backwards difference estimate ∂∂t2φ2 '
φ(t)−2φ(t−∆t)+φ(t−2∆t)
∆t2 was used. Separating terms dependent on the unknown source strengths from those independent of them, the free surface boundary condition may be written as
φ(t) + ∆t2g∂φ(t)
∂y '2φ(t−∆t)−φ(t−2∆t) . (2.63) This assumes at least two previous time steps have been solved. Otherwise for the first time step (t = 0) the condition φ(t) = 0 for t ≤0 substituted into (2.63) yields ∂φ∂y(0) = 0 on the free surface, and for the second time step (t= ∆t) ∂2φ∂t(∆2t) '
6φ(∆t)
∆t2 . The latter is based on the assumptionsφ(0) = ∂φ∂t(0) =∂2∂tφ(0)2 = 0 (the third of these being a consequence of the backwards difference operator applied toφ(t) = 0 fort≤0), which, with the known value ofφ(∆t), allows us to write φ as a cubic function of t, and the result follows by differentiation. In reality the backwards difference operator breaks down at the instant t = 0 because the fluid may receive a finite impulse. In fact differentiating the equivalent time domain Green function expression (equation (3.1)) analytically with respect to time, one obtains a non-zero real part in the second
15This assumes a rigid body. The more general case of a deforming body will be dealt with in the next chapter,
but using a Green function method. The simple source equivalent may be recovered from the Green function method by omitting the wave term (convolution integral) in the Green function, leaving only the body and image terms. There is also in this case a difference in sign of the image term.
derivative of the convolution integral term at t = 0 of Re n
igQ π(z−c¯)
o
= πgQImc (unless the body accelerates from rest, in which case Q(0) = 0). This of course contradicts the assumption
∂φ
∂y = 0. However many tests were carried out using different alternatives for ∂2φ(t)
∂t2 for the first and second time steps (t= 0, ∆t) using both the simple source and Green function methods. It was found in all cases, even with inconsistent approximations, that as ∆t →0 the final result was independent of the choice. This was not unexpected since the third and subsequent time steps all have consistent approximations for ∂∂t2φ2.
The equations equivalent to (2.63) for the first and second time steps are respectively ∂φ(0) ∂y = 0 (2.64) φ(∆t) +∆t 2g 6 ∂φ(∆t) ∂y ' 0 (2.65)
The solution to (2.64) is trivial since ∂φ∂y = −2Q implies that all the free surface source strengths must be zero. Only the body equations need to be solved for the first time step.
Returning to subsequent time steps, it follows from equations (2.63) and (2.65), recalling that ³ ∂φ ∂y ´ i= −Qi 2 , that (AF B)ij = (AF F)ij= φij−δ 2∆t2g t≥2∆t φij− δ 12∆t2g t= ∆t (2.66) where δ= 0 if i6=j 1 if i=j , and that (RF)i= 2φi(t−∆t)−φi(t−2∆t) t≥2∆t φi(0) t= ∆t . (2.67)
In equation (2.66) φij is analogous to ˜Bij in equation (2.59) without the wave terms and with the body and image sources having the same sign. However we note that the collocation point,z, will always be on the free surface, andcin the lower half plane, preventing the path of integration of ln (z−c) from crossing its branch cut on the negative real axis, and making the e−iγ term unnecessary. Also the difference in sign of the image term, not to mention its absence from the free surface sources, requires the term (c1−c2), as described in section 3.3.1, to be
retained. Thus φij = 1 2πRe © e−iβ((c 1−c2) + (z−c1) ln (z−c1)−(z−c2) ln (z−c2)) +eiβ((¯c 1−¯c2) + (z−¯c1) ln (z−¯c1)−(z−¯c2) ln (z−¯c2)) ª (2.68) in (AF B) (j≤m) or φij = 1 2πRe © e−iβ((c 1−c2) + (z−c1) ln (z−c1)−(z−c2) ln (z−c2)) ª (2.69) in (AF F) (j≥m+ 1).
Finally,φi(t−∆t) in equation (2.67) is evaluated using φi= n X j=1 φijQj (2.70)
whereφij andQj are the values corresponding to the previous time step, and should be known already, whileφi(t−2∆t) has already been evaluated asφi(t−∆t) in the previous time step.
Non-reflecting boundary condition at the free surface edges
This difficulty has been discussed in section 2.3.4 and is particularly relevant to time domain problems.
In the present work the difficulty was eventually ignored. Some non-reflecting boundary conditions were attempted without sufficient success to justify the additional complexity to the program. Yeung [101] (p411) comments “What then is the appropriate boundary condition that will have minimal effect on the interior solution? This has always been a ‘sore point’ in compu- tational fluid mechanics. There is no absolutely satisfactory answer to this nontrivial difficulty and the search still continues [references given].” while Raven [82] claims “But, particularly for 3D problems, no satisfactory non-reflective boundary conditions are known.” Although various authors have worked on the problem (for example [13], [8]), it is clearly a major undertaking to deal with the problem properly, and it was therefore decided to circumvent the problem by choosing to panel a sufficiently large section of the free surface, depending on the length of time to be simulated. Nevertheless there were reflections, but, provided the simulation time was limited, not sufficient to contaminate significantly the flow around the body. Of greater concern was the general drift of the free surface observed after large time simulations. This problem appears to initiate when the first waves reach the boundary edges and are reflected, so it is presumed to be a related problem and perhaps due in part to the lack of a mechanism for the dissipation of energy. It too was minimised by matching the size of the free surface domain to the length of time being simulated.
Calculation of wave height
Wave height can very conveniently be calculated using ∂η∂t = ∂φ∂y = −2Q on the free surface, from which (assuming an initially flat free surface) the free surface elevation at paneli is
ηi(t) =
Z t
0
−Qi(τ)
2 dτ. (2.71)
This may be evaluated givenQiat discrete time steps, using for example a trapezoidal rule. Alternatively, one can use
η =−1
g ∂φ
∂t, (2.72)
which appears to require more computational time, but, considering that the solution requires the calculation ofφanyway, the difference is negligible. The results from the two methods are
virtually indistinguishable (except in the case of a very large number of small time steps, when the former may suffer from accumulated roundoff error), and the choice is usually a matter of preference.