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SHIP GEOMETRY

RESISTANCE

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This section is concerned with fundamental aspects of the resistance of a ship, or body, to motion through calm water without consideration on the effects of the method of propulsion.

See: Wave, angle of diverging.

A correction made to the results of a hydrodynamic experiments made in a channel or tunnel of one cross-section in order to estimate the equivalent results for another cross-section. Specifically a correction made to the results of a resistance experiment in a towing tank in other to estimate the equivalent results in unrestricted water.

The region of fluid close to a solid body where, due to viscosity, transverse gradients of velocity are large as compared with longitudinal variations, and shear stress is significant. The boundary layer may be laminar, turbulent, or transitional. See also Flow, regime.

<m>{delta}, {delta^2} or {d_1}, {theta},{theta^*} or ????</m>

da aggiungere delta elevato **

- Boundary layer thickness (δ995): The dis-tance normal to the surface of a body at which the speed attains that in an equivalent inviscid flow. For practical purposes this is sometimes taken as 99.5% of the inviscid flow speed or 99.5% of the total head.
- Displacement thickness (δ
^{*}, δ_{1}: the dis-tance normal to the surface of a body by which streamlines outside the boundary layer are displaced. For two-dimensional flow:

<m>delta^* = int{y=0}{delta}{(1- overline{U}/U_delta)dy}</m>

where *U _{δ}* = the velocity at the edge of the boundary layer and U = velocity in the boundary layer.

- Momentum thickness (
*θ*): A parameter such that the quantity <m>phi bigcup{0}{2}{} theta</m>

is the defect in the rate transport of momentum due to the boundary layer. For two dimensional flow: <m>theta = int{y=0}{delta}{overline{U}/U_delta(1-overline{U}/U_delta)dy}</m> - Energy thichness <m>(delta^* ????)</m>

AGGIUNGERE DELTA DOPPIO *

A parameter such that quantity <m>1/2 phi bigcup{0}{3}{theta}</m>

is the defect in the rate of transport of ketic eny due to the boundaryyis
inerg laer. This given by:
<m>theta^* = int{y=0}{delta}{overline{U}/U_delta(1- overline{U^2}/U^2_delta)dy}</m>

A source-sink pair where the axial spacing tends to zero as the product of axial spacing and the source strength remains constant. The value of that product is the “moment” of the doublet, and the direction from the sink to the source is the “axis” of the doublet. Consequently, a doublet of moment M (dimension L^{4}T^{-1}) and of axis x located in a point A generates at any point P a velocity potential:

<m>phi = - M / {4pi r^2} {partial r}/{partial x}= - M/{4pi r^2} cos(theta)</m>

Where *r* = AP and *θ* = angle between AP and axis x^{(1)}. If M< 0, the axis of the doublet would be in the negative x-direction. In two dimensional problems, the definition holds. But the potential generate by a double of moment *M* (dimension
L^{3}T^{-1}) and of axis x is:

<m>phi = - M / {2pi r} {partial r}/{partial x}= - M/{2pi r} cos(theta)</m>

where *r* = AP and *θ* = angle between AP and axis x.

^{(1)}

See: Potential function or Velocity potential.

(*D*) [LMT^{-2}]
The fluid force acting on a moving body in such a way as to oppose its motion; the component of the fluid forces parallel to the axis of motion of a body. Drag is the preferred term in aerodynamics and for submerged hydrodynamic bodies, while resistance is generally used in ship hydrodynamics. The various forms of drag are defined in relation to resistance. See also Resistance.

(*C*_{D})[-]

The non-dimensional ratio of the drag per unit of a representative area of a body to the dynamic pressure far ahead of the body.

See Pressure, dynamic.

The flow of a viscous liquid in which layers of laminae of fluid appear to slide smoothly past each other. Momentum transfer and shear between neighbouring layers of fluid are due to molecular interactions only.

A flow field in which the fluid velocity <m> overline {U} </m>

is equal to the gradient of a scalar velocity potential <m>phi,overline{U} = grad phi i.e. </m>

in which no vorticity is present, curl <m>overline{U} = 0</m>.

See also Potential function.

A term referring to the state of the flow in any region; the principal recognised regimes are laminar, transitional, turbulent and separated flows.

Flow occurring in an eddy or separated zone in which the local flow has a component opposite in direction to that of the main flow.

A transverse flow induced by the boundary layer geometry and by pressure conditions existing in the main flow.

The detachment of the main fluid flow from a solid surface due to an adverse longitudinal pressure gradient sometimes caused by a sudden change of the direction or the curvature of the surface. The fluid in the separated flow contains eddies, and may be nearly static or may contain a region of reversed flow.

A flow in which there are rapid and apparently random fluctuations both in the magnitude and in the direction of velocity. The velocity fluctuations may also be described by a random spectrum of vortices of varying size and strength. Turbulent resistance is higher than that in laminar flow at the same Reynolds number, because of the high momentum exchange by transverse fluctuations.

The flow of a fluid where the flow characteristics include the effects of the shear forces acting on the fluid, and within it.

See: Resistance.

(*Fr*)[-]
A dimensionless parameter expressing the conditions of dynamical similarity for flow systems influenced by gravity and inertia alone. In particular it defines the speed at which geometrically similar models and ship will develop wave systems which are geometrically similar. It is given by:

*Fr* = <m>V/sqrt{gL}</m>

The length term L is usually the length of the ship. Other forms of the Froude number use some other characteristic dimension, such as the cube root of volume of displacement, the submergence depth or the depth of water in restricted waterways.

See Flow, potential.

See Sublayer, laminar.

See Equipotential line.

See Froude number.

See Reynolds number.

See Flow potential.

(φ) [L^{2} T^{-1}]

In irrotational motion of a fluid, the velocity at any point may be derived from a single function φ such that its derivative with respect to distance in any direction is equal to the velocity component in that direction.
See also Flow, potential.

(q) [L^{-1}MT^{-2}]
The pressure change corresponding to the reduction of the momentum of a fluid element to zero, <m>q = 1/2 rho U^2</m>

(p) [L^{-1}MT^{-2}]

The static pressure, *p*, at a point in a stream flow is that which would be recorded by a pressure gauge advancing with the speed of the local fluid and thus static with respect to it.

(*R*) [LMT^{-2}]

The fluid force acting on a moving body in such a way as to oppose its motion; the component of the fluid forces acting parallel to the axis of motion of a body. Resistance is the preferred term in ship hydrodynamics, while drag is generally used in aerodynamics and for submerged bodies. Total resistance is denoted by RT and various (not mutually exclusive) components of resistance are defined below. See also Drag.

(C_{F}, C_{R}, C_{S}, C_{T}, C_{V}, C_{W}, etc.)[-]

The non dimensional ratio of any specific component of resistance per unit area, to the dynamic pressure far ahead of the body.

(R_{F}) [LMT^{-2}]

The component of resistance obtained by integrating the tangential stresses over the surface of a body, in the direction of motion.

(C_{F})[-]

An alternative name for the coefficient of frictional resistance, in which the reference area is taken to be the wetted area under consideration.

(R_{P}) [LMT^{-2}]

The component of resistance obtained by integrating the normal stresses over the surface of a body in the direction of motion.

(R_{R}) [LMT^{-2}]

A quantity obtained by subtracting from the total resistance of a hull, a calculated friction resistance obtained by any specific formulation.

(R_{S}) [LMT^{-2}]

The component of resistance associated with the expenditure of energy in generating spray.

(R_{V}) [LMT^{-2}]

The component of resistance associated with the expenditure of energy in viscous effects.

(R_{PV}) [LMT^{-2}]

The component of resistance obtained by integrating the components of the normal stresses due to viscosity and turbulence. This quantity cannot be directly measured except for a fully submerged body when it is equal to the pressure resistance (R_{P}).

(R_{WP}) [LMT^{-2}]

A resistance component deduced from measurements of wave elevations remote from ship or model where it is assumed that the sub surface velocity field, and hence the momentum of the fluid, can be related to the wave pattern by means of linearised theory . The resistance so deduced does not include wave-breaking resistance.

(R^{W}) [LMT^{-2}]

The component of resistance associated with the expenditure of energy in generating gravity waves.

(*Re*)[-]
A dimensionless parameter expressing the condition of dynamical similarity for flow systems influenced by viscosity and inertia alone. For equal values of Reynolds number and the same orientation to the flow, the specific resistance coefficients of all geometrically similar smooth surfaces are identical as long as the uninfluenced speed field are similar and the flow is influenced by viscosity and inertia alone.
It is given by:

<m> Re = {VL rho} / mu = VL / nu </m>

The length term L is usually the length of the surface, but the distance from the leading edge of the surface to a specific point, the diameter of a body, or the thickness of the boundary layer are sometimes used as length terms.

<m>(tau)</m> [L^{-1}MT^{-2}]
In a viscous fluid, the shear stress is the tangential resisting force per unit area acting on any boundary within the fluid. The specific value of the shear stress at a wall is denoted by <m>tau</m>_{w}.

A point at which fluid is assumed to be with drawn symmetrically from all directions. The velocity potential due to a sink has the same form as the potential due to a source, but the strength *Q* is negative. See also Source.

A point from which fluid is assumed to flow symmetrically in all directions. The strength Q of a source is defined in a three dimensional flow as the volume of fluid issuing in unit time; its dimensions are L^{3}T^{-1}. (Some authors use <m>sigma = Q/{4pi}</m>volume flow as source strength). A source at a point A generates at any point P a velocity potential due to such a source of strength *Q* is:

<m>phi = -Q/{4 pi r}</m>

where *r* = AP.
In a two dimensional flow parallel to a plane, a source at a point A is in fact a uniform distribution of sources on a straight line passing through A normal to the plane. The velocity potential due to such a source of strength *Q* is:

<m>phi = Q/{2pi} lnr</m>

where *r* = AP and ln = natural logarithm.
*Q* is the volume of fluid issuing per unit time and per unit length in the direction normal to the plane. The dimension of *Q* is L^{2} T^{-1}. An irrotational flow of perfect fluid may be represented as due to distributions of source and sinks, or doublets, on some set of points.

A Kelvin source is defined by the potential generated by a constant source in uniform rectilinear motion below the free surface of a perfect fluid.

(in high speed craft) [LT^{-1}]
The speed at which the resistance reaches a maximum before a planing craft enters the planing phase, or a hydrofoil craft enters the foilborne phase.

A line in a fluid such that its tangent at any point is parallel to the instantaneous velocity of the fluid at that point.

A very thin layer of laminar flow, within a turbulent boundary layer and adjacent to a solid surface.

(μ) [L^{-1}MT^{-1}]
The quantity expressing the resistance of a fluid to internal shear; the ratio of tangential stress to rate of shear deformation in flow of an incompressible Newtonian fluid. For unidirectional shear flow:

<m>(nu)</m> [L^{-1}MT^{-1}]

The ratio of the coefficient of dynamic viscosity to the mass density of the fluid:
<m>nu = mu/rho</m>

See also Liquid Properties and Physical Constants.