Resistance Primer

The resistance primer covers the decomposition of forces. The coverage of the primer will include:

  1. Scaling
  2. Viscous Resistance
  3. Wavemaking Resistance

Scaling

To accurately predict forces, the model and ship need dynamic similarity. There are three parts to dynamic similarity: scaling of model sizes, model speeds, and model forces.[1]

Geometric similarity is the linear scaling of the model to the ship (Eqn(1)). This is met when all characteristic lengths of the model (Lmodel) are made proportional to the ship (Lship) by the scaling factor (lambda).[1]

L_ship / L_model = lambda (1)

Kinematic similarity is the scaling of speeds (Eqn(2)).[1] Kinematic similarity requires that the ratio of velocities of the model (Vmodel) and ship (Vship) are the same.[3]

V_ship / V_model = sqrt(L_ship / L_model) = sqrt(lambda) (2)

Dynamic similarity is the scaling of forces (Eqn(3)). In the tank, hull resistance is the force of the water acting on the ship. All forces (F) can be decomposed into their non-dimensional coefficient (C).[3]

C = F / (1/2 rho S V^2) (3)

Besides force, a coefficient is dependent on the mass density of the fluid (r), the velocity (V) of the fluid relative to the characteristic length (L) of the body, and the characteristic wetted surface area of the hull (S). Total resistance coefficient (CT) is the summation of viscous, wavemaking, air, and added wave resistances (Eqn(4)). The primary resistances are viscous (CV) and wavemaking (CW), and the others are omitted for simplicity.[3]

Ct = Cw + Cv (4)

Viscous Resistance

The viscous resistance coefficient (CV) is a function of Reynolds number (Re) and the frictional resistance coefficient (Cf). Reynolds number defines the speed of the vessel relative to the condition of fluid flow. It is the ratio of inertia forces to viscous forces (Eqn(5)).[1]

Re = V L / nu (5)

Reynolds number is dependent on the velocity of the fluid, the characteristic distance of fluid flow over the body (L), and the kinematic viscosity of the water (nu).  There is one modification to the Reynolds number calculation which is exclusive to sailing craft. For normal non-sailing vessels, the flow over the body (L) is typically the length of the waterline. For sailing craft, the keel and rudder make a large difference in frictional resistance. Moreover, the local Reynolds number for both keel and rudder are much less than the Reynolds number across the entire hull. Therefore, the flow conditions across much of the wetted surface area for sailing craft is at a lesser Reynolds number than if the whole body was the characteristic flow length. The result is an approximation to Reynolds number (Rn) for sailboats: a factor of 0.7 is added to the formula for Reynolds number (Eqn(6)).

Rn = 0.7 V L / nu (6)

The frictional resistance coefficient (Cf) can be closely approximated without modeling by using the standard 1957 International Towing Tank Conference (ITTC) ship-model correlation equation (Eqn(7)). The frictional resistance equation is a function created through the curve fits of many years of experimental data.[3]

C_f = \frac{0.075}{{\left( \log_{10} R_n - 2 \right)}^2} (7)

The viscous resistance equation includes the skin friction term and a form factor (k). The form factor accounts for added resistance from the shape of the vessel. This reduces to an equation for the viscous resistance coefficient (Eqn(8)).[3]

C_V = \left(1+k \right) \cdot C_f (8)

Wavemaking Resistance

The wavemaking resistance coefficient (CW) is a function of the non-dimensional Froude number (Fn). Froude number defines the speed of the vessel relative to the type of wave systems created. It is the ratio of inertial forces to gravity forces (Eqn(9)).[1]

Fn = \frac{V}{\sqrt{g L}} (9)

Froude number is dependent on the velocity of the fluid, the acceleration due to gravity (g), and the characteristic distance of fluid flow over the body.  There is no equation for the wavemaking resistance coefficient, and it remains a function of only Froude number (Eqn(10)).[2]

C_W = f \left( Fn \right) (10)

Since it is assumed that only viscous and wavemaking resistances are significant, and there exists an equation for the viscous resistance coefficient, an equation of resistance can be defined in Eqn(11). Wavemaking becomes the only unknown variable in the resistance equation and is found experimentally through tank testing or CFD.[3]

R_T = \frac{1}{2} \rho S V^2 \left( 1 + k \right) \left( \frac{0.075}{\left( \log_{10} Rn - 2 \right)^2} + C_W \right) (11)

References

1.  Franzini, Joseph B. and E. John Finnemore. Fluid Mechanics with Engineering Applications.  9th Ed.  WCB/McGraw-Hill.  Boston, MA.  1997.

2.  Tupper, Eric. Introduction to Naval Architecture.  Society of Naval Architects and Marine Engineers.  Jersey City, NJ.  2000.

3.  Van Manen, J.D. and P. Van Oossanen. "Resistance."   Principles of Naval Architecture.  Society of Naval Architects and Marine Engineers.  Jersey City, NJ.  1988.

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