42 SUBMERGED BODY IN WAVES 174 42 Submerged Body in Waves A fixed, submerged body has a circular cross-section of constant diameter one meter, with length twenty meters. For the calculations below, use the Morison formulation for wave-induced force, with Cd =1.2. Fix the wave amplitude at one meter, and consider the frequency range of 0.1 to two radians per second. For each of the first four items below, in addition to making a plot please answer: What is the maximum magnitude and at what frequency does it occur? What are the physical mechanisms that create the curve you see? 1. Consider the body oriented horizontally and pointing into waves, at a fixed depth of four meters. Calculate and show in a graph the total heave force magnitude as a function of wave frequency ω. See Figure 1. The maximum magnitude of vertical force is 18.1kN, at about 1.1 rad/s. Since the wave amplitude is taken as constant at all frequencies, the Morison forces are increasing rapidly as the frequency goes up. At about 1.75 rad/s, however, there is a zero net force because the wavelength is the same as the body length. The fact that this plot is made for a unity wave amplitude means that the square of this function could be multiplied by the wave spectrum, to get the spectrum of the force. Beware of the deep-water assumption however, as indicated below. 2. Under the same conditions, calculate and show in a graph the total pitch moment magnitude as a function of wave frequency. Take the moment around the center of the body (L/2). The maximum pitch moment magnitude is 122 kNm at 1.45 rad/s. Explanation is similar to the case of vertical force, except that the zero frequency is now quite high; it is where the wavelength is equal to about sixty percent of the body length (not half!). 3. Now let the body be vertically oriented - e.g., it is a fixed piling. Using the deep-water wave formulas, calculate and show what is the magnitude of the total surge force on the piling, as a function of wave frequency. The maximum surge force magnitude seen is 16.7kN, at the maximum considered frequency of two radians per second. The forces get larger and larger because the velocities and accelerations at all depths both grow with frequency. This effect is stronger than the effect of the depth envelope getting narrower, over most of the frequency range. As above, also note that the wave amplitude is taken to be one meter for all frequencies. 4. Under the same conditions, calculate and show in a graph the pitch moment magnitude as a function of wave frequency. Take the moment around the lower end of the body. The maximum pitch moment magnitude seen is 292kNm, at the maximum considered frequency of two radians per second.� � � 42 SUBMERGED BODY IN WAVES 175 5. If the water depth is just twenty meters, for what range of frequencies is the analysis of the vertical body loading reasonable? In deep water, we approximate the wave velocity as g/k, whereas in the general case it is g tanh(kh)/k,where h is the water depth. Figure 2 shows that tanh(kh) is close to one for all frequencies above one radian per second. Hence all of the maxima reported above are valid because they occur above this value, typical to SS 3. In practice, we have to design for higher sea states, which have lower frequency content as well as larger amplitudes; in these cases, we have to take into account the shallow-water formulae. 0 0.5 1 1.5 2 0 5 10 15 20 Horizontal Body: Heave Force, kN 0 0.5 1 1.5 2 0 5 10 Vertical Body: Surge Force, kN Horizontal Body: Pitch Moment around Center, kNm Vertical Body: Pitch Moment around Lower End, kNm 0 0.5 1 1.5 2 0 50 100 150 rad/s 0 0.5 1 1.5 2 0 50 100 150 rad/s Figure 9: Force and moment on the horizontally-oriented body (left), and the vertically-oriented body (right).42 SUBMERGED BODY IN WAVES 176 sqrt(tanh(20*k)): Discrepancy in Wave Velocity 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 frequency, rad/s Figure 10: The deep-water approximation is reasonable when this function is near one. %----------------------------------------------------------% Forces on a submerged body in waves clear all; d = 1 ; % cylinder diameter, m L = 20.0 ; % cylinder length, m Cd = 1.2 ; % Morison drag coefficient T = 4 ; % depth of centerline, when the body is horizontal, m dx = L/50 ; % differential length for strip theory calculations g = 9.81 ; % gravity, m/s^2 rho = 1000 ; % water density, kg/m^3 A = 1 ; % wave amplitude, m ntimes = 41 ; % number of time steps per cycle to study wvec = .05:.05:2 ; % frequency vector, rad/s xvec = dx/2:dx:L-dx/2 ; % body longitudinal location vector; centers for i = 1:length(wvec) ; % loop through the frequencies of interest w = wvec(i) ; k = w^2/g ; % wave number tvec = 0:2*pi/w/(ntimes-1):2*pi/w ; % make up a one-cycle time % vector for this freq. for jj = 1:length(tvec), % loop through time t = tvec(jj) ; for j = 1:length(xvec) ; % loop through longitudinal locations x = dx*j ;42 SUBMERGED BODY IN WAVES 177 % first get the vertical velocity, acceleration, and % Morison force % on the horizontal body; note the Archimedes force! e1 = exp(-k*T); v1 = A*e1*(-w)*sin(w*t-k*x) ; % local vertical velocity vv1 = A*e1*(-w^2)*cos(w*t-k*x) ; % local vertical acc. dFz(j) = 1/2*rho*Cd*d*dx*v1*abs(v1) + ... rho*(pi/4*d^2)*2*dx*vv1 ; % differential force % now get the horizontal velocity, acceleration, and % Morison force on the vertical body; note the % Archimedes force! e2 = exp(-k*(L-x+4)) ; % (move from the seafloor to the surface) v2 = A*e2*w*cos(w*t) ; % local horiz. veolcity (horiz. position is zero, % so kx=0) vv2 = A*e2*(-w^2)*sin(w*t) ; % local horiz. acceleration dFx(j) = 1/2*rho*Cd*d*dx*v2*abs(v2) + ... rho*(pi/4*d^2)*2*dx*vv2 ; end; Fz(jj) = sum(dFz); % net vertical force on horiz. body Momz(jj) = sum(dFz.*(xvec-L/2)); % pitch moment around horiz. body mid-point Fx(jj) = sum(dFx) ; % net horizontal force on vertical body Momx(jj) = sum(dFx.*xvec); % pitch moment around lower end of vert. body end; Fzf(i) = max(abs(Fz)) ; % get the
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