Cavity Shape

The cavity shape that develops behind the foil fore-body is characterised as for the fence/ramp cavity flows described in Chapter 3: that is, in terms of length and thickness parallel and perpendicular to the free stream direction respectively and volume, as shown in Figure 4.1b. The general shape of the cavity surface is convex when viewed from the liquid. This is a necessary condition for a cavity shape to be valid here, as the pressure in the cavity is assumed to be the minimum in potential flow theory (Brennen, 1995; Milne- Thomson, 1968). The cavity surface detaching from the edge of the step was found to be convex under all flow conditions examined. On the opposite trailing edge, under certain conditions, the cavity surface detached with an initial concave curvature and then after a distance of 0.5c or less behind the trailing edge it inflected to give a convex shape that was then maintained for the remaining length of the cavity surface. Any such cavity shape was considered invalid and not included in the data set.

A sample of results for the relationship between the non-dimensional cav- ity length, thickness and volume and σc for the t/c = 0.25%, h/c = 0.01%

foil profile are given in Figure 4.8. These results are typical for the range of foil thicknesses examined. The full set of results obtained is included in Appendix C for reference.

As discussed in Section 3.4.3 there is a power law relationship between cavity length and σc for finite length supercavities, expressed by:

lc

c =Aσ

n

c (4.5)

For a symmetric body in an infinite flow field withσc ≪1 the linearised theory

predicts that the exponent n = ₋2. For the intercepted foils in this present study, that are not symmetric due to the addition of the step on one side, this power law relationship still holds as can be seen in Figure 4.8a . The value of the exponent n is increased slightly above that from linearised theory and is found to be invariant with γ but to increase slightly with body thickness, as shown in Figure 4.9a. There is also a similar trend of an increase in n with fence height. The effect of varying γ is shown in Figure 4.8a, with the offset of the curves corresponding to the change in coefficient A of Equation 4.5. This latter relation is a linear function of γ, as shown in Figure 4.9b. For a foil with a 25% t/c and 1% h/c at zero incidence, this relationship is A = 0.0174γ◦

+ 0.0877. This results in a 10% increment in the dimensionless cavity lengthlc/c per degree change inγ for a constant σc.

The relationship between cavity volume and σc also follows a power law,

but deviates over the shorter cavity lengths with the curves bending slightly off the straight line for σc >0.1 as shown in Figure 4.8b. For cavity thickness

(Figure 4.8c) there is again a general power law trend with a similar deviation to that of the volume data over the upper end of the σc range.

1 10 100 0.01 0.1 1 lc /c σ_c 2.5 3.0 3.5 4.0 5.0

(a) Dimensionless cavity length as a func- tion ofσc. 0.01 0.1 1 10 0.01 0.1 1 Vol/c 3 σ_c 2.5 3.0 3.5 4.0 5.0

(b) Dimensionless cavity volume as a func- tion ofσc. 0.1 1 10 0.01 0.1 1 tc /c σc γ (º) 2.5 3.0 3.5 4.0 5.0 6.0 7.0

(c) Dimensionless cavity thickness as a function ofσc.

Figure 4.8: Variation of dimensionless cavity geometric properties with σc.

(withα= 0◦

,t/c= 25%,h/c = 1% andrmax withγ a parameter). These rela-

tionships are well represented by a power law function ofσc (see Equation 4.5)

withγ not affecting the slope but giving a constant offset to the curves. In the cavity length case (a), it equates to a 10% increment inlc/c per degree change

-2.0 -1.9

0 0.1 0.2 0.3

n

t/c

(a) Exponentnas a function of t/c.

0 0.1 0.2 3 4 5 6 7 A γ (º) (b) CoefficientAas a function ofγ. Figure 4.9: From Equation 4.5 (lc/c = Aσcn) typical results showing: (a) the

relation betweennand t/c(forα = 0◦

,h/c = 1%), which is invariant toγ; (b) the linear relation between A and γ (with α = 0◦

, t/c = 25% and h/c = 1% gives A= 0.0174γ◦

+ 0.0877).

The sensitivity of cavity shape to the shape of the step attached to the foil trailing edge is illustrated in Figure 4.10 for two different step profiles with the same flow. Note that the two step shapes examined represent the two extremes of a gentle ramp and a perpendicular forward-facing step, and as such should be indicative of the upper and lower bounds of the effect of the step shape. From the results given in Chapter 3 it was found that for the wedge flows studied, the cavity shape was insensitive to the shape of the wedge fore-body and depended solely on the slope of the trailing edge, i.e. the detachment angle of the cavity. The ramp used hasβ = 60◦

and the fenceβ = 90◦

giving a change inβ of 30◦

between the two shapes. Figure 4.10 shows that the cavity shape is relatively insensitive to such a large change in the shape of the step, with a small increase in cavity size over the full range of σc. The offset of the

cavity length versus σc curve (Figure 4.10a) equates to a 6% reduction from

the fence to the ramp results, which is a small but not insignifcant difference due only to the change in step shape.

0.1 1 10 100 0.01 0.1 1 lc /c σ_c

(a) Dimensionless cavity length as a func- tion ofσc. 0.1 1 10 100 0.01 0.1 1 Vol/c 3 σ_c

(b) Dimensionless cavity volume as a func- tion ofσc. 0.1 1 10 0.01 0.1 1 tc /c σc γ (º) Fence Ramp

(c) Dimensionless cavity thickness as a function ofσc.

Figure 4.10: Effect of the step shape (fence or ramp) on cavity geometry as a function of σc. (with α = 2.5◦, t/c= 25%, h/c = 1%, γ = 6◦, β = 90◦ - fence

and β = 60◦

- ramp). There is an equivalent reduction in cavity size with the change from the fence to the ramp for allσc. In (a) this offset equates to a 6%