5.2 Computation of Viscous Drag on Appendages Computation of Viscous Drag on Appendages
Appendages on a fin-keeled sailing vessel include the keel fin and the rudder, Appendages on a fin-keeled sailing vessel include the keel fin and the rudder, and possibly a ballast bulb and winglets on the keel or rudder, or both (for and possibly a ballast bulb and winglets on the keel or rudder, or both (for an exam
an example showple showing such appending such appendageages, s, see Figursee Figure e 14 later in 14 later in the articthe article)le). . InIn principle, the viscous drag on appendages could be computed either with a principle, the viscous drag on appendages could be computed either with a RANS method or with a method that strongly couples an outer inviscid flow RANS method or with a method that strongly couples an outer inviscid flow with inner boundary layer equations.
with inner boundary layer equations.
With the existing state of the art, the drag forces provided by RANS codes With the existing state of the art, the drag forces provided by RANS codes hav
have not had e not had the accurathe accuracy needed for the design of racing vesselcy needed for the design of racing vessels. s. DevDevelop- elop-ment of outer inviscid solutions strongly coupled with inner integral boundary ment of outer inviscid solutions strongly coupled with inner integral boundary layer equations is in the formative stage for three dimensional flows and shows layer equations is in the formative stage for three dimensional flows and shows consid
considerable proerable promise for the futuremise for the future. . On the other hand, the strongly coupleOn the other hand, the strongly coupledd method for two-dimensional (2D) flows is very advanced and shows excellent method for two-dimensional (2D) flows is very advanced and shows excellent agreement with experiments. It can be used in support of design, because the agreement with experiments. It can be used in support of design, because the rudder and keel
rudder and keel fin and fin and optional wingleoptional winglets of ts of a high-performa high-performance sailing vesseance sailing vessell are high-aspect–ratio lifting surfaces, so their friction drag can be estimated are high-aspect–ratio lifting surfaces, so their friction drag can be estimated from 2D section analysis.
from 2D section analysis.
A review of the coupled method in two and three dimensions is given here.
A review of the coupled method in two and three dimensions is given here.
The flow away from the immediate vicinity of the lifting surface is largely The flow away from the immediate vicinity of the lifting surface is largely in-vi
viscscidid, b, butut visviscocousus efeffefectctss araree imimporportatantnt inin thethe bouboundandaryry lalayeyersrs. . FoForr mamanyny yeyeararss researchers tried to iterate between the inviscid and boundary layer solutions.
researchers tried to iterate between the inviscid and boundary layer solutions.
The idea was to compute an inviscid flow, use its pressure gradients in solving The idea was to compute an inviscid flow, use its pressure gradients in solving the integral boundary layer equations, solve again for the inviscid flow with the the integral boundary layer equations, solve again for the inviscid flow with the
0
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n is the normal into the lifting surface,is the normal into the lifting surface, dsds is the differential element of length oris the differential element of length or area (2- or 3D),
is aa ficfictittitioiousus trtrananspspiriratationion sosourcurcee ststrerengtngthh disdistrtribibutiutionon onon ththee liliftftinging susurfrfacacee anandd wake that has to be determined so as to make the outer flow the same as the real wake that has to be determined so as to make the outer flow the same as the real boundary layer would cause. The solution to Equation 8 with the last term boundary layer would cause. The solution to Equation 8 with the last term re-mo
moveved,d, anandd susubjebjectct toto ththee ususuauall NeNeumumanannn bouboundandaryry cocondindititionon anandd ththee KuKuttattacocon- n-dition, is called the inviscid potential,
dition, is called the inviscid potential, φφinvinv. The total velocity potential is then. The total velocity potential is then
where GGvv is the sum of is the sum of GG and a and a body-shabody-shape-spepe-specific dipole distributiocific dipole distribution n on on thethe surface, chosen such that the normal derivative of
surface, chosen such that the normal derivative of GGvv is zero except where theis zero except where the source and field points coincide.
source and field points coincide.
TWO-DIMENSIONAL FLOW
TWO-DIMENSIONAL FLOW The surface velocity, which corresponds to the tan-The surface velocity, which corresponds to the tan-gential velocity at the outer edge of the boundary layer, is called
gential velocity at the outer edge of the boundary layer, is called U U ee and isand is obtained as the derivative of the total potential with respect to the tangential obtained as the derivative of the total potential with respect to the tangential coordinate,
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SAILING VESSEL DESIGN
SAILING VESSEL DESIGN 627627 dissipation coefficient, and
dissipation coefficient, and τ τ is the shear stress andis the shear stress and uu is the local velocity in theis the local velocity in the boundary layer.
boundary layer.
Drela & Giles (1987) give the semi-empirical equations for all the above Drela & Giles (1987) give the semi-empirical equations for all the above boundary layer parameters in terms of
boundary layer parameters in terms of θ θ ,, M M , and, and U U ee for laminar flow, whichfor laminar flow, which is then entirely specified by these parameters and the simultaneous solution is then entirely specified by these parameters and the simultaneous solution of Equation
of Equations 10, 12, and 13. s 10, 12, and 13. The user must eiThe user must either specther specify the locatiify the location of theon of the transition point from a laminar to a turbulent boundary layer or use a transition point from a laminar to a turbulent boundary layer or use a semi-empir
empirical relatiical relation to on to estimestimate where natural transate where natural transition occurs. ition occurs. One commonOne common me
meththodod isis babasesedd onon anan esestitimamatete ofof ththee raratitioo ofof ththee amamplplititududee ofof ththee momostst ununststabablele T
Tollollmimienen-S-Schchlilichchtitingng wawaveve atat ththee trtranansisititionon popointint toto itsits vavaluluee atat thethe firfirstst loclocatatioionn of growth with the ratio expressed as
of growth with the ratio expressed as eenn˜˜. The value of . The value of ˜˜nn at transition has beenat transition has been correlated with
correlated with T T f f , the ratio of root mean square–free stream turbulence speed, the ratio of root mean square–free stream turbulence speed to mean speed, by Mack (1977) as
to mean speed, by Mack (1977) as
˜˜
n
n == −−((88..4343 ++ 22..4 log4 logT T f f ).). (14)(14) Drela & Giles (1987) give a semi-empirical function,
Drela & Giles (1987) give a semi-empirical function, f f 11, for the , for the rate of changerate of change of
of ˜˜nn along the chordalong the chord d
d nn˜˜
d
dss −− f f 11(( H H ,, θθ )) == 00.. (15)(15) For turbulent boundary
For turbulent boundary layerlayers, s, they provide semi-empthey provide semi-empiricairical l relatrelations for ions for allall of the boundary layer parameters in terms of
of the boundary layer parameters in terms of θ θ ,, M M ,, U U ee, and the coefficient, and the coefficient of maximum shear stress in the boundary layer,
of maximum shear stress in the boundary layer, C C τ τ , for which they provide a, for which they provide a
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628
628 MILGRAMMILGRAM
Figure 6
Figure 6 Calculated and measured pressure coefficients on an airfoil section. TheCalculated and measured pressure coefficients on an airfoil section. The upper part upper part of of the
the figufigurere shoshowsws thethe prepressussurere coecoeffifficiciententss witwithh thethe ininvisviscidcid calcalculculatationion ((dasdashedhed linlinee)) anandd ththee cocoupupleledd boundary layer calculation (
boundary layer calculation (solid linesolid line). The). Thelower part lower part of the figure shows the airfoil section andof the figure shows the airfoil section and the streamlines at the outer edge of the boundary layer.
the streamlines at the outer edge of the boundary layer.
edge, there is very substantial boundary layer thickening on the suction side edge, there is very substantial boundary layer thickening on the suction side and reduced thickness on the pressure side. This is shown by deviations from and reduced thickness on the pressure side. This is shown by deviations from
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SAILING VESSEL DESIGN
SAILING VESSEL DESIGN 629629
m m w w w w w w . . a a n n n n u u a l a l r r e e v v i i e e w w s s o o r r g g . .
3 3 / / 1 1 1 1 . . F F o o r r p p e e r r s s o o n n a a l l u u s s e e o o n n l l y y . .
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630
630 MILGRAMMILGRAM
m m w w w w w w . . a a n n n n u u a l a l r r e e v v i i e e w w s s o o r r g g . .
3 3 / / 1 1 1 1 . . F F o o r r p p e e r r s s o o n n a a l l u u s s e e o o n n l l y y . .
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SAILING VESSEL DESIGN
SAILING VESSEL DESIGN 631631 giv
given by en by Mughal (1992) are used Mughal (1992) are used by both by both MileMilewski and Nishida.wski and Nishida.
∂ s-sions for these thicknesses in terms of the velocity components through the sions for these thicknesses in terms of the velocity components through the
m
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632
632 MILGRAMMILGRAM
m m w w w w w w . . a a n n n n u u a l a l r r e e v v i i e e w w s s o o r r g g . .
3 3 / / 1 1 1 1 . . F F o o r r p p e e r r s s o o n n a a l l u u s s e e o o n n l l y y . .
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