13.42 LECTURE 13: FLUID FORCES ON BODIES. Using a two dimensional cylinder within a two-dimensional flow we can demonstrate some of the principles
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1 13.42 LECTURE 13: FLUID FORCES ON BODIES SPRING 2003 c A. H. TECHET & M.S. TRIANTAFYLLOU 1. Morrison s Equation Using a two dimensional cylinder within a two-dimensional flow we can demonstrate some of the principles we will use to determine forces on bodies in unsteady flow. Fluid particles in contact with a body surface are motionless relative to the body so that a boundary layer is formed. From basic fluid mechanics we know that for an inviscid flow around a cylinder we have two stagnation points at which the pressure is maximum. In the case of a viscous flow the fluid particles follow the sides of the cylinder but as they pass a point on the cylinder, where the pressure gradient becomes adverse, the particles will begin to separate. This separating creates the wake which is constrained within the boundary (shear) layer originating from the separation point. In a turbulent flow the fluid in the boundary layer has stronger mixing, allowing the vorticity to spread out and diffuse and for the flow to remain attached. Thus the separation point is further aft and the wake is narrower and pressure (form) drag is reduced from the laminar case. The condition of the cylinder surface (roughness) may also cause the early transition to turbulent flow, however in this case the frictional drag increases due to roughness. Following the above discussion we can think of the drag as influenced by the width of the wake (narrower wake implies less profile drag for the same diameter object) and can better understand the discussion from lecture 12 using the system energy to determine the drag force on a body. From last lecture the resulting force on the body takes the form of Morrison s Equation. This is a combination of an inertial term and a drag term Spring
2 2 SPRING 2003 c A. H. TECHET & M.S. TRIANTAFYLLOU (1.1) F x (t) = D(t) = ρc M U ρc D A U U In order to obtain rough estimates it is advantageous to use Morrison s equation with constant coefficients. Suppose we want to find the estimates of the wave forces on a fixed structure. The procedure is as follows: 1.) Select an appropriate wave theory (linearized waves, or other higher order if necessary). 2.) Select the appropriate C M and C D based on Reynolds number, and other factors. 3.) Apply Morrison s Equation Wave Theory C D C M Comments Reference Linear Theory Mean values for ocean wave data on 13-24in cylinders Wiegel, et al (1957) Recommended design values based on statistical analysis of published data Agerschou and Edens (1965) Stokes 3rd order Mean Values for oscillatory flow for 2-3 in. cylinders Keulegan and Carpenter (1958) Stokes 5th order Recommended values based on statistical analysis of published data Agerschou and Edens (1965) We can see from the above table that for linear waves the recommended values for drag and mass coefficients are and 2.0 respectively. The range of drag coefficients allows us to account for roughness and Reynolds number effects. These values are for rough estimates. In reality these coefficients vary widely with the carious flow parameters and with time. Bretschneider showed that the values of C D and C M can even vary over one wave cycle. Even if we ignore the time dependence of these coefficients we must account for the influence of other parameters.
3 FLUID FORCES ON BODIES 3 Reynolds number and roughness: For smooth cylinders at Reynolds numbers around 10 5 laminar flow transitions to turbulent flow, and there is a dip in C D as a function of Re. For larger Reynolds numbers the separation point remains essentially constant and thus so does the drag coefficient. In this range C D is Reynolds number independent. Roughness causes the change from laminar to turbulent flow at a lower Reynolds number and increases the friction and causes a larger C D. The mass coefficient is influenced by the changes in the boundary layer and is thus also effected Reynolds number and roughness. z l u(z,t) 0 x Figure 1. Cylinder in non-uniform inflow Suppose a vertical cylinder is subject to a current with a horizontal velocity changing both in time and vertically in the z-direction: u(z, t). The approach in practice is to evaluate using Morrison s formula the force per unit length (eq. 1.2) at each point along the cylinder length and then integrate to get the total force (eq. 1.3). (1.2) F(z,t) = C M ρ π 1 4 d2 u(z,t) + C D ρ d u(z,t) u(z,t) 2 (1.3) F(t) = l 0 F(z,t) dz
4 4 SPRING 2003 c A. H. TECHET & M.S. TRIANTAFYLLOU The moment on the structure around the origin (point 0) is found as (1.4) M(t) = l 0 z F(z,t) dz There are several limitations to these integrals. First we are limited to assume that each section does not influence the adjacent sections flow. This assumption becomes questionable in the case of a cross flow that introduced a direct relation among the flow of neighboring sections. The second assumption is that the cylinder is not piercing the free surface because in this case the water splashing must be taken into account. In the absence of experimental data we use Morrison s equation as a first estimate, but we should attempt a correction for the surface piercing phenomema. 2. Forces on an Inclined Cylinder Suppose that a cylinder of diameter, d, and large length, l, is at an angle within an unsteady inflow, u(t), and we would like to use Morrison s equation. It has been suggested that in such cases of slender objects (large ratio l/d) that we can use the following approach. First decompose the inflow velocity into two components U n and U t where U n is the velocity normal to the cylinder and U t is the component tangential to the cylinder. The we can use the following expressions: (2.1) F n = C M ρ π 4 d2 U n + C D 1 2 ρ d U n U n (2.2) F t = C f 1 2 ρ πd U t U t to determine the normal and tangential forces per unit length. The mass and drag coefficients are found using the diameter of the cylinder and the normal velocity as in the general (non-inclined) case. The frictional coefficient, C f, is used in the tangential case instead of the drag coefficient since the drag results from the flow along (tangential) to the cylinder.
5 FLUID FORCES ON BODIES 5 In the three-dimensional case where we have a vertical cylidner subject to a velocity vector (u, v, w) then we must decompose this velocity into components normal and tangential to the cylinder. The convention is to take the z-axis parallel with the cylinder axis (centerline) so that the w component is the tangential component of velocity: U n = u 2 + v 2 and U t = w. Then the above equations (2.1 and 2.2) can be used. These equations have limitations. Experiments have shown that these expressions are valid up to incline angles of about Morrison s Equation When both the body and fluid are moving Assume that a vertical cylinder is moving with velocity u(t) within a fluid with velocity v(t), both velocities in the horizonal direction and uniform in space, then we can write Morrison s equation as follows: (3.1) F(t) = 1 2 ρ C D d h [v(t) u(t)] v(t) u(t) + ρ C M π 4 d2 h v(t) ρ (C M 1) π 4 d2 h u(t) where d is the cylinder diameter and h the cylinder height. It is good to note that this equation does not account for the inertial force due to the mass of the cylinder as required by Newton s law. For example if the cylinder was left in the free stream the we would set F(t) = m u(t), where F(t) is from equation 3.1 and m is the mass of the cylinder. 4. Recap: Relative Importance of Inertia versus Drag Force 4.1. Integrated Force. If we consider the force caused on a vertical cylinder in depth H by a linear wave with wavelength, λ and height h then the inertial force F I is: (4.1) F I = 0 H C M π 4 d2 ρ h 2 ω2 cosh[k(z + H)] sinh(kh) cos(ωt) dz The wave elevation is (4.2) η(x,t) = h 2 cos(kx ωt)
6 6 SPRING 2003 c A. H. TECHET & M.S. TRIANTAFYLLOU By integrating we find the inertial force (4.3) F I = π 4 ρ d2 ω 2 h 2k cos(ωt) Similarly we can find the drag force F D (4.4) (4.5) F D (t) = 0 H 1 2 ρ C D d ω 2 h2 4 = 1 2 ρc D d ω 2 h2 4 cosh 2 [k(z + H)] sinh 2 (kh) sin(ωt) sin(ωt) dz sin(ωt) sin(ωt) sinh(2kh) + 2kH 4k sinh 2 (kh) The maximum values of these two force components is found to be: (4.6) F IM = π 4 ρ d2 ω 2 h 2k (4.7) F DM = 1 2 ρ C D d ω 2 h2 4 sinh(2kh) + 2kH 4k sinh 2 (kh) To compare the effects of inertial versus drag components we can take the ratio F I to F D. (4.8) F D = C D h F I πc m d sinh(2kh) + 2kH 4 sinh 2 (kh) Let s take the typical values for drag and mass coefficients, 1.0 and 2.0 respectively. We can thus find how the maximum drag force compares to the maximum inertial force: F DM > F IM when h d > 2π 4 sinh 2 (kh) sinh(2kh)+2kh 5. Inertia Dominated Waves In the regimes where drag is unimportant (below 10% of the total force) we use the theory of inertia dominated flows neglecting the viscous terms. When a large object is moving with stead speed near the water free surface then it is subject to a steady force that is caused by skin friction, separation, and wave making. The last part can be treated as an inviscid flow (wave making resistance).
7 FLUID FORCES ON BODIES 7 For this section let s concentrate on the problem of forces and motions caused by waves on large objects. We will denote the incident wave potential by φ I e iωt. These are the waves with frequency ω as if no structure existed. We denote the total potential by φ which satisfies the following relations: (1) Inside the fluid Laplace s equation holds: 2 φ = 0 (2) On the bottom there is no normal velocity: φ n = 0 (3) On the structure φ n = V n, where V n is the velocity normal to the structure surface. (4) At the free surface, the kinematic and dynamic boundary conditions must be satisfied. These equations are nonlinear and the general solution to the resulting problem is in general impossible. For this reason we use the first order combined boundary condition: φ tt + g φ z = 0 at z = 0. (5) At infinity we must use an appropriate radiation condition. Such that the potential φ φ I must contain only outgoing waves. Otherwise we are left with a meaningless solution of waves generated at infinity. As a result of these assumptions the problem becomes linear. It is to be expected, however, that the error will be huge for cases of extreme waves and structure motion and we must correct for this when designing. In general though for irregular ocean waves the present approach provides useful results. Since the problem is linear we can then write the total potential as a sum of the incident, diffraction, and radiation potentials. (5.1) φ = (φ I + φ D + φ R ) e iomegat The diffraction potential, φ D, is caused by the presence of the structure when it is not moving, and the radiation potential is caused by the body motions. The body, unconstrained in any fashion, is assumed to be free to move in six-degrees of freedom: surge, sway, heave, roll, pitch, and yaw. These motions are denoted by x j where j = 1,2,3,4,5,6. It is customary
8 8 SPRING 2003 c A. H. TECHET & M.S. TRIANTAFYLLOU to write the complex potential in the following form: (5.2) φ R = 6 j=1 ẋ j φ j where ẋ j is the velocity in the jth direction and φ j is the potential caused by a unit motion in the jth direction. This potential decomposition is the fundamental step in vessel motion prediction and can be summarized as follows: The total potential, φ, describing the fluid flow around a structure in the presence of waves, equals the sum of the undisturbed wave potential, φ I, the potential due to the presence of the body when it is motionless, φ d, and the radiation potential, φ R caused by the body motion. This will be the fundamental building block for our analysis of ship motions in waves.
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