Eighteenth Symposium on Naval Hydrodynamics (1991) / Chapter Skim
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Three-Dimensional Instability Modes of the Wake Far Behind a Ship
Three-Dimensional Instability Modes of the Wake Far Behind a Ship
Pages 553-566
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From page 553... ...
It is shown that the complex phase velocity of the unstable waves satisfies Howard's semi-circle theorem. For a self-similar velocity profile, a numerical solution is obtained by expanding the perturbation pressure in a Fourier series and solving a set of simultaneous ordinary differential equations.
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From page 554... ...
The incompressibility condition requires that the perturbation velocity has to be divergence-free: aU+vv = 0 ax (2) At the free surface we have the kinematic and dynamic conditions for the free-surface elevation q(x,y,t)
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From page 555... ...
= 0 (12) Since unstable waves have complex phase velocity, U- c does not vanish anywhere in the flow field, and the integral in (12)
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From page 556... ...
Thus, given that the free surface elevation is proportional to the value of the dynamic perturbation pressure there, Mode I disturbs the free surface in an antisymmetric manner around y = 0, whereas Mode II disturbs the free surface in a symmetric manner (figure 3)
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From page 557... ...
Consequently, the new variable q can be expanded in a cosine Fourier series containing odd terms only. The Fourier series is twice differentiable with respect to ~ because of the boundary conditions (30)
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From page 558... ...
satisfies the boundary conditions (30) , and can be expanded in a cosine Fourier series: q(r,[~)
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From page 559... ...
. F=1.5 E-2 _ OFT 559 n= Figure 4c: Variation of the amplitude of the Fourier coefficients n=1,3,5,7,9 with r for the most unstable wave F=1.5 (semi-log scale)
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From page 560... ...
The convergence rate is therefore algebraic, and we can safely say that the main advantage in using the Fourier series expansion, as opposed to a direct numerical solution of (10) , lies in the wide separation in magintude between the first and the subsequent Fourier coefficients, which allows an accurate representation of the series using very few coefficients.
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From page 561... ...
-plane yields frequencies with positive imaginary parts, the flow is unstable. This was done in the previous subsection, where an unstable wavenumber range was found, by solving the dispersion relation with respect to the frequency.
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From page 562... ...
The free surface elevation thus consists of two parallel series of alternating hills and valleys the height of which increases exponentially with x. The wave grows indefinitely according to linear theory, but in reality the growth will be eventually saturated by non-linear effects, unless the free surface wave breaks before that.
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From page 563... ...
An interesting observation is that, at low Froude numbers, the frequency and phase velocity of the instability waves is controlled by the characteristics of the shear flow in the wake, and is practically the same as in the F=0 case. This allows the derivation of a very simple approximation for the unstable modes, as follows: We first solve for the eigenvalue co and eigenvector q'(r)
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From page 564... ...
Fluid Mech., 190,p.419.
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From page 565... ...
In fact, from Howard's theorem, it is straightforward to see that in the limit as the growthrate of the unstable wave tends to zero the phase velocity has to remain within the range of the basic flow. Consequently, I do not see how in the similar problem that the discusser has considered that the phase velocity of neutrally stable modes can possibly lie outside this range.
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