Bug fix in fowt.calcQTF_slenderBody()

We were missing the transverse force at the ends of the cylinder from Rainey's formulation

Author: lucas-carmo

raft/raft_fowt.py

@@ -1575,14 +1575,13 @@ def calcQTF_slenderBody(self, waveHeadInd, Xi0=None, verbose=False, iCase=None,
                         f_rslb  += - rho*v_i * aux
 
 
-                        # ----- axial/end effects  ------
+                        # ----- end effects  ------                               
                         # note : v_i and a_i work out to zero for non-tapered sections or non-end sections
+                        # TODO: Should probably skip this part if along the length of the element, i.e. if a_i and v_i are different from zero due to tapering
                         if circ:
                             v_i = np.pi/12.0 * abs((mem.ds[il]+mem.drs[il])**3 - (mem.ds[il]-mem.drs[il])**3)  # volume assigned to this end surface
-                            a_i = np.pi*mem.ds[il] * mem.drs[il]   # signed end area (positive facing down) = mean diameter of strip * radius change of strip
                         else:
                             v_i = np.pi/12.0 * ((np.mean(mem.ds[il]+mem.drs[il]))**3 - (np.mean(mem.ds[il]-mem.drs[il]))**3)    # so far just using sphere eqn and taking mean of side lengths as d
-                            a_i = (mem.ds[il,0]+mem.drs[il,0])*(mem.ds[il,1]+mem.drs[il,1]) - (mem.ds[il,0]-mem.drs[il,0])*(mem.ds[il,1]-mem.drs[il,1])
 
                         f_2ndPot += mem.a_i[il]*p_2nd*mem.q # 2nd order pressure
                         f_2ndPot += rho*v_i*Ca_End*np.matmul(mem.qMat, acc_2ndPot) # 2nd order axial acceleration
@@ -1593,6 +1592,11 @@ def calcQTF_slenderBody(self, waveHeadInd, Xi0=None, verbose=False, iCase=None,
                         p_drop    = -2*0.25*0.5*rho*np.dot(np.matmul(mem.p1Mat + mem.p2Mat, u[:,i1, il]-nodeV[:, i1, il]), np.conj(np.matmul(Ca_p1*mem.p1Mat + Ca_p2*mem.p2Mat, u[:,i2, il]-nodeV[:, i2, il])))
                         f_conv   += mem.a_i[il]*p_drop*mem.q
 
+                        u1_aux    = np.matmul(Ca_p1*mem.p1Mat + Ca_p2*mem.p2Mat, u1_aux)
+                        u2_aux    = np.matmul(Ca_p1*mem.p1Mat + Ca_p2*mem.p2Mat, u2_aux)
+                        f_transv  = 0.25*mem.a_i[il]*rho*(np.conj(u1_aux)*nodeV_axial_rel[i2, il] + u2_aux*np.conj(nodeV_axial_rel[i1, il]))
+                        f_conv   += f_transv
+
                         F_2ndPot += translateForce3to6DOF(f_2ndPot, mem.r[il,:])
                         F_conv   += translateForce3to6DOF(f_conv, mem.r[il,:])                   
                         F_axdv   += translateForce3to6DOF(f_axdv, mem.r[il,:])