7 Theoretical
7.9 The finite wing
Theoretical methods – basic concepts 159 on the geometric thickness to chord ratio of the section.
Therefore, these earlier relationships should be used with some caution, since the lift slope and zero lift angle correction factors are governed by the growth of the boundary layer over the aerofoil to a significant degree.
With the increasing use of computational fluid dynamics in propeller and ship flow analysis problems a number of turbulence models are encountered. For example, these might include:
• Thek–ε model in either the standard or Chen and Kim extended model.
• Thek–ωmodel in the standard, Wilcox modified or Menter’s baseline model.
• Menter’s one equation model.
• The RNGk–εturbulence model.
• Reynolds stress models.
• Menter’s SSTk–ωturbulence model.
• The Splalart–Allmaras turbulence model.
In the case of thek–εmodel it was found that if the generic turbulent kinetic energy equation were coupled to either a turbulence dissipation or turbulence length scale modelling equation, then it gave improved per- formance. The energy and dissipation equations as formulated by Jones and Launder (Reference 23) rely on five empirical constants, one of which controls the eddy viscosity and two others, which are effectively Prandtl numbers, which relate the eddy diffusion to the momentum eddy viscosity. Sadly these constants are not universal constants for all flow regimes but when combined with the continuity and momentum equations form the basis of thek–εmodel for the analysis of tur- bulent shear flows. Thek–ω model is not dissimilar in its formulation to thek–εapproach, but instead of being based on a two equation approach its formula- tion is centred on four equations. The Reynolds stress models, frequently called second-order closure, form a rather higher level approach than either thek–εor k–ω approaches in that they model the Reynolds stresses in the flow field. In these models the eddy vis- cosity and velocity gradient approaches are discarded and the Reynolds stresses are computed directly by either an algebraic stress model or a differential equa- tion for the rate of change of stress. Such approaches are computationally intensive but, in general, the best Reynolds stress models yield superior results for complex flows, particularly where separation and reat- tachment are involved. Moreover, even for attached boundary layers the Reynolds stress models surpass the k–εmodel results and it is likely that they will become dominant in the future.
The boundary layer contributes two distinct com- ponents to the aerofoil drag. These are the pressure drag (Dp) and the skin friction drag (Df). The pres- sure drag, sometimes referred to as the form drag, is the component of force, measured in the drag direc- tion, due to the integral of the pressure distribution
over the aerofoil. If the aerofoil were working in an inviscid fluid, then this integral would be zero – this is d’Alembert’s well-known paradox. However, in the case of a real fluid the pressure distribution decreases from the inviscid prediction in the regions of separated flow and consequently gives rise to non-zero values of the integral. The skin friction drag, in contrast, is the com- ponent of the integral of the shear stressesτwover the aerofoil surfaces, again measured in the drag direction.
Hence the viscous drag of a two-dimensional aerofoil is given by:
2D viscous drag=skin friction drag+pressure drag that is,
Dv=Df+Dp (7.51)
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160 Marine propellers and propulsion
Figure 7.24 Flow over a finite aspect ratio wing: (a) plan view of blade; (b) flow at blade tip and (c) schematic view of wing tip vortices
Figure 7.25 Downwash distribution for a pair of tip vortices on a finite wing
Theoretical methods – basic concepts 161
Figure 7.26 Derivation of induced drag
shown in Figure 7.25. This distribution derives from the relationship
downwash at any pointy=contribution from the left-hand vortex +contribution from the right-hand vortex that is,
ω(y)= − 4π
1
(b/2+y)+ 1 (b/2−y)
ω(y)= − 4π
b (b/2)2−y2
(7.53)
where the span of the aerofoil is b. The downwash velocityω(y) combines with the incident free stream velocityV to produce a local velocity which is inclined to the free stream velocity at the blade section, as shown in Figure 7.26, by an angleαi.
Consequently, although the aerofoil is inclined at a geometric angle of attackαto the free stream flow, the section is experiencing a smaller angle of attackαeff
such that
αeff =α−αi (7.54)
Since the local lift force is by definition perpendic- ular to the incident flow, it is inclined at an angleαito the direction of the incident flow. Therefore, there is a component of this lift forceDiwhich acts parallel to the free stream’s flow, and this is termed the ‘induced drag’
of the section. This component is directly related to the lift force and not to the viscous behaviour of the fluid.
As a consequence, we can note that the total drag on the section of an aerofoil of finite span comprises three distinct components, as opposed to the two components of equation (7.51) for the two-dimensional aerofoil, as follows:
total 3D drag=skin friction drag+pressure drag
+induced drag (7.55)
that is,
D=Dv+Df+Di
where the skin friction dragDf and the pressure drag andDpare viscous contributions to the drag force.
In Figure 7.24 it was seen that a divergence of the flow on pressure and suction surfaces took place. At the trail- ing edge, where these streams combine, the difference in spanwise velocity will cause the fluid at this point to roll up into a number of small streamwise vortices which are distributed along the entire span of the wing, as indicated by Figure 7.27. From this figure it is seen that the velocities on the upper and lower surfaces can be resolved into a spanwise component (v) and an axial component (u). It is the difference in spanwise compon- ents on the upper and lower surfaces,vUandvLrespect- ively, that give rise to the shed vorticity as sketched in the inset to the diagram. Although vorticity is shed along the entire length of the blade these small vortices roll up into two large vortices just inboard of the wings or blade tips and at some distance from the trailing edge as shown in Figure 7.28. The strength of these two vortices will of course, by Helmholtz’s theorem, be equal to the strength of the vortex replacing the wing itself and will trail downstream to join the starting vortex, Figure 7.12;
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162 Marine propellers and propulsion
Figure 7.27 Formation of trailing vortices
Figure 7.28 Schematic roll-up of trailing vortices
again in order to satisfy Helmholtz’s theorems.
Furthermore, it is of interest to compare the trailing vortex pattern of Figure 7.28 with that of Prandtl’s clas- sical model, shown in Figure 7.1, where roll-up of the free vortices was not considered. Although Figure 7.28 relates to a wing section, it is clear that the same hydro- dynamic mechanism applies to a propeller blade form.