earised problems
Calculation of body surface pressures are important in the evaluation of lift and drag forces on translating bodies, or for the evaluation of hydrodynamic coefficients for oscillating bodies.
A question arises as to whether it is consistent to calculate the exact pressure on the body boundary, given that the free surface boundary condition has been linearised, or whether the body pressure calculation should also be linearised in some sense. This is not a trivial question, and its answer can differ depending on the problem. It is also a question that has concerned various authors. Wehausen [97] (pp. 141-42) has the following to say:
An approximation scheme in which some of the equations of the problem are sat- isfied to a higher order of approximation than others is said to be inconsistent...[Two examples follow.]
Neither [of the two examples] is convincing and in principle each inconsistent problem is still only of first order accuracy. There is, in fact, no guarantee that the change from the consistent first order problem is in the direction of greater accuracy... One particularly misleading version of an inconsistent problem is one in which a flow is generated by a distribution of sources and sinks selected to satisfy the linearised free-surface condition and from which a “body” is generated by tracing streamlines numerically.
The problem has also been considered by Tuck [90], who says his results “show that, by a factor of at least 2 or 3 in the range of Froude numbers where wave-making is significant, it is more important to correct for non-linearity at the free surface than for the fact that the body
boundary condition is not satisfied exactly by the first approximation of the body surface.” Tuck’s paper however does not deal with some crucial points, and this will be discussed below after a more complete description of the problem.
Yeung [101] (p. 399) allows application of the exact body boundary condition for a deeply submerged body, but states “If the body is not deeply submerged, a consistent linearisation procedure requires that condition (2.3) [his body boundary condition] onH[the body surface] be satisfied in some linearised manner also.”
To illustrate the difficulty let us look in particular at the calculation of pressures on a body in steady translation beneath a linearised free surface.
In section 2.3.2 it was stated that linearisation of the free surface in two dimensional flows rests on one of two assumptions: small motions, or deep submergence. It is clear that the small motions assumption can not be applied to a translating body, particularly at steady state where infinite time, hence infinitely large motions, are implicit. This leaves deep submergence as being the only valid justification for linearising the free surface boundary condition.
It is extremely important to recognise the correct form of the free surface linearisation if the body surface is also to be linearised without misleading results. The difference between the two assumptions above will be highlighted in the following discussion, and will be shown in a specific case to amount to a factor of 2.
Consider first the small motion assumption, applicable to some other classes of problem. We wish to calculate pressures using Bernoulli’s equation. It is sufficient to consider only those terms that differ from hydrostatic pressure if we are interested in hydrodynamic forces, hence from equation (2.9) we note that dynamic pressure is
p=−ρ µ ∂φ ∂t + 1 2∇φ· ∇φ ¶ . (2.21)
The flow viewed from the moving body is assumed to be at steady state, thus ∂φ∂t +U∂φ∂x = 0 and p=−ρ µ −U∂φ ∂x + 1 2∇φ· ∇φ ¶ . (2.22)
Ignoring second order terms inφgives
p'ρU∂φ
∂x. (2.23)
Consider next the case ofdeep submergence, applicable to the present problem of a translating body. We will assume the flow to be a perturbation of the flow about a translating body in an infinite fluid (the limiting case for infinitely deep submergence). Let the potentialφbe composed of the potential produced by the same body in an infinite fluid, φ∞, and a perturbation, φP,
i.e. φ=φ∞+φP. Thenp=−ρ(∂φ∂t∞ +∂φ∂tP +12(∇φ∞· ∇φ∞+ 2∇φ∞· ∇φP+∇φP· ∇φP)) =
−ρ(−U(∂φ∞ ∂x +
∂φP
∂x ) + 12(∇φ∞· ∇φ∞+ 2∇φ∞· ∇φP +∇φP · ∇φP)). Noting that the non-
circulating potential flow about any body in an infinite medium gives rise to no net force, then if the purpose is to calculate net forces the terms involving only φ∞ can be ignored. Further,
it follows from the deep submergence assumption that the perturbation potential φP is small, therefore the second order term∇φP· ∇φP can also be ignored. Thus
p'ρ µ U∂φP ∂x − ∇φ∞· ∇φP ¶ .
This expression may be simplified by noting that φand φ∞ satisfy on the body boundary
∇φ·ˆn=∇φ∞·nˆ=U nx, from which it followsφP satisfies∇φP·nˆ = 0. If the body boundary
makes an angleθ with the x-axis in the anticlockwise direction (i.e. nx=−sinθ, ny = cosθ) then the boundary condition for φP gives us ∂φP
∂y = ∂φP
∂x tanθ. If we further define α such that ∂φ∞
∂x = αU, then ∂φ∞
∂y =U(α−1) tanθ. Finally making these substitutions, and noting ¡
1 + tan2θ¢ = 1
cos2θ, we can write the deep submergence linearisation equivalent of equation (2.23), p'ρU∂φP ∂x µ 1−α cos2θ ¶ . (2.24)
Returning to the small motion assumption we can assist the comparison by writingφ=φ∞+
φP, but in this case we assume onlyφ, andnot φP, to be small. However, as in equation (2.24), we can ignore the contribution fromφ∞in equation (2.23) since it makes no net contribution to
the total hydrodynamic force acting on the body, hencep'ρU∂φP
∂x . We see then that the deep submergence assumption results in an increase in hydrodynamic force by a factor of¡1−α
cos2θ ¢
over the small motion assumption.
In order to illustrate how significant the factor¡1−α
cos2θ ¢
is, consider now the specific case of a circular cylinder in translation. The analytic solution toφ∞ is well known (e.g. Havelock [36],
p82), and gives us that
u∞ = ∂φ∞
∂x =−Ucos 2θ v∞ =
∂φ∞
∂y =−Usin 2θ on the cylinder boundary, from whichα=−cos 2θ, hence
µ 1−α cos2θ
¶ = 2.
The wrong choice of body linearisation will in this case result in a factor of 2 error. Clearly if a linearised form of the body pressure calculation is to be used care must be taken to choose its form correctly.
It is evident from the above that to evaluateonlythe first order term in thecorrectlylinearised pressure for a deeply submerged body it is necessary to distinguish the contribution ofφP from that ofφ∞. This is impractical for the general problem since it involves solving forφ∞as well as
φ, and the question arises whether (as implied by Wehausen [97] and Tuck [90]) it is necessary. If the problem is a valid candidate for linearisation (i.e. the body is submerged deeply enough that the linearised free surface boundary condition closely approximates the exact one) it should not matter whether non-linear terms on the body surface are included or not. Wehausen does not contradict this but does state that inclusion of the terms does not imply any improvement
in accuracy beyond first order. We want however assurance that inclusion of the terms does not result in degradation of accuracy.
To see whether inclusion of the ∇φP · ∇φP term adversely affects results, assume the total potentialφnow to satisfy thenth order free surface boundary condition, and can be written as φ =φ∞+ Pn i=1φ (i) P where φ (1)
P is identical to φP above and φ
(i)
P is the ith order term of the potential. The exact free surface boundary condition is satisfied as n→ ∞. If εrepresents the linearisation parameter then we noteφ(Pi)=O¡εi¢on the free surface, butO¡εi+1¢on the body
surface10, whileφ
∞=O(ε) on the free surface butO(1) on the body surface. Then the leading
terms in∇φ· ∇φare∇φ∞· ∇φ∞+∇φ∞· ∇φ(1)P +∇φ∞· ∇φ(2)P +∇φ
(1)
P · ∇φ
(1)
P +. . ., which are of order 1,ε2,ε3,andε4 respectively on the body surface. We see that, in agreement with Tuck
[90], the effect on the body of non-linearity at the free surface ³
∇φ∞· ∇φ(2)P ´
is more significant than non-linearity in the body pressure calculation
³
∇φ(1)P · ∇φ(1)P ´
, but further conclude that the latter is of high enough order that its inclusion or not is irrelevant.
The crucial qualification in the previous paragraph is “if the problem is a valid candidate for linearisation”. One measure of this is the wave steepness, orH/LwhereHis the wave height for regular waves (2×amplitude) andL is the wavelength, which can be applied to the essentially regular train of waves that appears behind any steadily translating body in two dimensions. It is well known that as H
L increases the wave peak sharpens and the trough flattens until eventually a cusp forms at the peak and the wave breaks. Michell [68] and Havelock [40] have shown that the limiting value of H
L is '0.142. Presumably then if any linear theory predicts a wave steepness approaching 0.142 or greater then it has well exceeded its range of validity.
Returning now to Tuck [90], his commentary on figure 1 of his paper (which illustrates his proposition with a particular example where the free-surface streamline bears no resemblance to the linearised free surface) includes “presumably the exact non-linear solution would involve highly non-sinusoidal or even breaking waves”. His linearised free surface has a steepness H
L = 0.541 = 3.8סH
L ¢
max so it is well beyond the limit where the exact solution even exists11. It
can also be shown that the limiting value of H
L (based on the linear estimate ofH) is exceeded for the entire Froude number range in his figure 3 (in which he shows the wave resistance force for a circular cylinder whose axis is submerged by one diameter) and most of figure 4 (showing wave lift for the same body), therefore the results presented in those figures are not necessarily meaningful. In his final figure the wave steepness in the central portion of the Froude number range is now just within the limiting value, and as expected his first order, inconsistent second order, and consistent second order theories are in this case in much better agreement.
The conclusion is that Tuck’s paper does demonstrate that the inconsistent second order calculation of forces on the body (i.e. without extending the free surface to second order) is
10The wave terms in equation (2.30) decay with depth, and the body is deeply submerged.
11Tuck does comment “This is a case in which we expect that the disturbance produced by the circle is too
severe for the potential (3.4) to be valid as the first term in a convergent series representing the exact potential,” but apparently fails to further recognise that the exact potential does not exist.
neither worse nor better that the first order calculations, but he does not demonstrate this for any physically possible flow. Also, his model, based on an asymptotic series of dipoles, differs from the present panel method solution in that only in the limiting cases of infinite order or of infinite depth does his solution satisfy the exact body boundary condition. It is perceived therefore that in the present method the effect of body non-linearity may be less.
It is without significant reservation therefore that pressures may be calculated in problems involving the deep submergence assumption in accordance with (2.22).
In problems involving the small motion assumption the more correct form is equation (2.23), but again it should not make any difference if equation (2.22) is used instead. Unlike the deep submergence problem however, it is far easier to use the linear form in this case. The linear form of pressure is also to be preferred in small motion problems because it is often the case that the motion is assumed to be of arbitrary (typically unit) amplitude, such as in the calculation of added mass and damping coefficients (see section 2.6.4), and the results scaled accordingly after the calculation. In such a case the inclusion of non-linear terms would be entirely erroneous.