21st Australasian Fluid Mechanics ConferenceAdelaide, Australia10-13 December 2018

Surface Pressure of Inclined Slender-Body Flows

S.-K. Lee and M. B. Jones

Defence Science and Technology Group, Melbourne, VIC 3207 AUSTRALIA

Abstract

This paper reports high-resolution surface-pressure maps ob-tained from inclined slender-body flows. The pressure maps areinterpreted with the aid of an empirical model developed froman extensive flow-visualisation database. This model predictsflow separation on the leeward side of arbitrary axisymmetricslender bodies. The present findings show that, on the leewardside, the flow separation is in the vicinity of the local maximain the time-averaged distribution of surface pressure.

Introduction

The surface flow over an inclined slender body produces verydistinct separation lines [2–4]. These characteristic lines areof interest because they relate to the off-body wakes and thepressure forces acting on the body. An understanding of theseparation lines and how they compare with the surface pressurecan be useful, for example, to improve the dynamic control ofslender-body type vehicles and to assist numerical modelling.

For the flow under consideration (figure 1), the separation linesare treated as transient, and the time scale is given by [2]:

t =x

U∞ cos(ψ), t∗ = t

U∞ sin(ψ)r(x)

=x

r(x)tan(ψ), (1)

where r(x) is the local body radius, ψ is the incidence angle andU∞ is the free-stream velocity. The advantage of the scaling (1)is that it can be applied to any arbitrary slender bodies.

A recent flow-visualisation experiment and review study [3]show that, for a large collection of slender-body shapes at in-cidence ψ up to 35 ◦, the scaling (1) provides a collapse of theleeward separation lines. These separation lines are the primaryand secondary negative bifurcations (B−1 and B−2 ) as shown, forexample, in figure 1; they fall on the power laws [3]:

ΘB−1= 151.2(t∗)−0.190, ΘB−2

= 137.5(t∗)0.045, t∗ ≥ 1.5. (2)

The azimuthal (Θ) locations of separations are indepen-dent of Reynolds number ReL = LU∞/ν over the range2.1×106 ≤ ReL ≤ 23×106 for a slender body of length L and

x=0

Ucos( )ψ

U sin( )ψ

U

x=L

x( )r

180

0

x

z

ψ

y

vorticesΘ Leeward

Plane normalto axisx

Surface area, A

Surface−flowseparation envelope

B−1

B−2

o

o

Figure 1. Flow separation on an inclined slender body in translation.

a working fluid of kinematic viscosity ν. In figure 1, the narrowregion of surface flow between B−1 and B−2 is known as the “sep-aration envelope”. The aim is to establish the time-averagedsurface pressure for comparison with the separation envelope.

Slender-Body Geometry and Experimental Technique

The geometry used here consists of a NACA0018 nose of lengthln/L = 0.23, a cylindrical midsection of length lm/L = 0.47 anda tapered tail of length lt/L = 0.30. It has a fineness ratio R =L/(2rm) = 7.3, a body length L = ln + lm + lt and the profile:

r(x)L

=

a0

√xL−a1

xL−a2

(xL

)2+

a3

(xL

)3−a4

(xL

)4,

0≤ x≤ ln,

12R

, ln < x≤ L− lt ,

12R− 1

2R

(ltL

)−2(xL−1+

ltL

)2, L− lt < x≤ L,

(3)

where a0 = 0.2322, a1 = 0.1126, a2 = 0.4097, a3 = 0.4321 anda4 = 0.2012 define the shape of the nose; see figure 2.

−0

.10

0.1

r(x

)/L

(a)

Nose Midsection

Truncation

Tail

Pressure-tapping locations

with max. radius rm

−0

.50

0.5

1

Cp

(b)

Increasing ψ

Increasing ψ ψ = 0ο

ψ = 16ο

Windward side, Θ = 0o

−0

.50

0.5

1

Cp

(c)

Increasing ψψ = 0ο

ψ = 16ο

Flank, Θ = 90o

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x/L

−0

.50

0.5

1

Cp Increasing ψ

(d)

Leeward side, Θ = 180o

ψ = 0ο

ψ = 16ο

Figure 2. (a) Body geometry and locations of surface-pressure tappings.(b)-(d) Longitudinal distributions of surface pressure Cp, Eq. (4), at se-lected Θ locations for ψ = 0 ◦ to 16 ◦ at incremental steps of 2 ◦.

(a) ψ = 2o

(c) ψ = 6o

(b) ψ = 4o

(e) ψ = 10o

(f) ψ = 12o

(d) ψ = 8o

(g) ψ = 14o

(h) ψ = 16o

(i) ψ = 16o; surface−flow pattern

B−2 B

−2

B−2B

−2B

−2

B−2

B−2

B−1

B−1B

−1B

−1

B−1 B

−1B

−1

Cp

Cp

Cp

Θ(d

eg.)

180

120

60

0

Θ(d

eg.)

180

120

60

0

Θ(d

eg.)

180

120

60

0

Low−pressurefront

Windwardstagnation

CpLocus of min.

Emergence ofbifurcation lines

x/L x/L x/L1.00 0.2 0.6 0.80.4 1.00 0.2 0.6 0.80.4 1.00 0.2 0.6 0.80.4

0.500.450.400.350.300.250.200.150.100.05

0−0.05−0.10−0.15−0.20−0.25−0.30−0.35−0.40−0.45−0.50

0.50

0.400.45

0.350.300.250.200.150.100.05

0−0.05−0.10−0.15−0.20−0.25−0.30−0.35−0.40−0.45−0.50

0.50

0.40

0.30

0.20

0.10

0

−0.10

−0.20

−0.30

−0.40

−0.50

Separation envelope

Leeward side

Windward side

0.35

0.25

0.15

0.05

−0.05

−0.15

−0.25

−0.35

−0.45

0.45

Figure 3. (a)-(h) Surface pressure Cp; the bifurcations B−1 and B−2 are modelled by Eqs. (1)-(3). (i) Example of a China-clay image for ψ = 16 ◦.

For testing in the low-speed wind tunnel at the Defence Scienceand Technology Group, a carbon-fibre composite body is man-ufactured with a length (L) of 2 m. The tapered tail is truncatedat x/L = 0.95 to allow mounting on an automated sting-columnrig. To monitor pitch and roll, an inclinometer (Jewel LCF-3000) is fitted inside the body; the standard errors are 0.013 ◦ inpitch and 0.046 ◦ in roll.

On each quadrant of the body, there is a row of 33 surface-pressure (ps) tappings and 1 additional tapping at the tip of thenose; see figure 2(a). Inside the body, the tappings are con-nected to a series of miniature differential pressure (electronicDTC) scanners with an operating range of 1 kPa up to 35 kPa.The surface-pressure signals are captured by a commercial dataacquisition system (MSS-8400) and are used to calculate thepressure coefficient:

Cp =ps− p∞

12 ρU2

∞

, (4)

where ρ is the density and p∞ is the static pressure of the free-stream. The wind-tunnel flow (U∞) is operated at 60 m/s with

ReL = 8×106. The pressure (ps) at each tapping is sampled at100 Hz and averaged over a period of 6 seconds. Pitch angles(ψ) are tested up to 16 ◦. To trip the flow, a circumferentialring of trip dots (diameter of 1.27 mm, thickness of 0.152 mm,and center-to-center spacing of 2.54 mm) is fixed at x/L = 0.05.The method of sizing the trip dots is from [1].

Surface-Pressure Distribution

Figures 2(b)-(d) provide the longitudinal distributions of pres-sure at different quadrants of the body. Inspection shows anincrease in ψ produces a higher Cp on the windward side of thenose (Θ = 0 ◦) and a lower Cp on the leeward side (Θ = 180 ◦).The Cp along the flank (Θ = 90 ◦) decreases with increasing ψ.

From a collection of longitudinal distributions of Cp such as infigures 2(b)-(d), it is possible to construct detailed maps of thesurface pressure as shown in figures 3(a)-(h). This is achievedexperimentally by rotating the axisymmetric body about its lon-gitudinal (x) axis at incremental steps ∆Θ = 4.5 ◦, where eachmap contains a total of 33×360 ◦/4.5 ◦ = 2640 pressure points.

0 4 8 12 16ψ (deg.)

00.0

04

0.0

08

|Cz|

Integrated Cp

Load balance

0 4 8 12 16ψ (deg.)

00.0

02

0.0

04

|Cm

|

(a) (b)

Figure 4. Comparison between integrated Cp and load-balance mea-surements; (a) side-force and (b) moment coefficients; the half errorbars show the uncertainties in ψ (pitch) due to wind-on deflection.

For ψ > 0 ◦, the Cp maps are folded about the flow-symmetryplane (in the xz direction; see figure 1) to minimise scatter. Forψ = 0 ◦, the map is axisymmetric and is not shown here.

Comparison with Load Measurements

To check that the measurement grid is sufficiently fine, the pres-sure is integrated over the surface (area A) of the body to provideload estimates for comparison with strain-gauge data. Figure 4shows, for example, a plot of the integrated Cp for pitch motionin the z direction (see figure 1); the side-force coefficient is

Cz =−1A

∫ L

0

∫ 2π

0Cp r sin(Θ) dΘdx (5)

and the moment coefficient about the nose (due to pitching) is

Cm =− 1AL

∫ L

0

∫ 2π

0Cp r

(x+

drdx

r)

sin(Θ) dΘdx. (6)

For each nominal value of ψ, a total of 22 samples (+) are ob-tained from an internal (6-component Aerotech) strain-gaugebalance; the scatter of the load-balance data in figure 4 is up to∼±0.250×10−3 for Cz and ∼±0.125×10−3 for Cm.

In figure 4, the measured body pitch angle (ψ) takes into ac-count the effect of wind-on deflection. For the largest angleψ = 16 ◦, the deflection (monitored by the inclinometer) is nomore than 1 ◦, and the maximum difference between measure-ments obtained from integrating Cp and from the load balanceis no more than 5% for Cz and 3% for Cm.

Comparison with Flow Visualisation

To assist interpretation, figure 3 includes the separation lines B−1and B−2 modelled by Eqs. (1)-(3). For validation, figure 3(i) pro-vides an example of a China-clay pattern to demonstrate flowseparation over the body at ψ = 16 ◦. The process required theclay mixture to be exposed to the flow for a few minutes to drybefore still photography. The 180 ◦ view in figure 3(i) is con-structed from longitudinal strips of the recorded images; this isachieved by rotating the body (at incremental steps of 5 ◦) aboutits x axis with a fixed position of the still camera. For furtherdetails on the China-clay technique, see [3].

Figure 3 shows the lines B−1 and B−2 for ψ>∼6 ◦; they extendupstream of the tapered tail with increasing ψ and they definethe separation envelope of the surface flow. The separation en-velope is located leeward of the locus of minimum Cp. On thenose, the maximum Cp of the windward stagnation is diametri-cally opposite to the low-pressure front (see figure 3). By takingcircumferential slices at selected x/L locations, figure 5 showsthat the separation envelope (•−•) modelled by Eqs. (1)-(3) is inthe vicinity of the local maxima or “plateaux” of Cp.

To further visualise the surface-pressure distribution, the Cp infigures 3(a)-(h) is replotted as a function of Eq. (1) in figures

−0.8

0

Cp

(a)

ψ = 0ο

16ο

x/L = 0.505

B1

− B

2

− Increasing ψ

at steps of 2o

Min. Cp

−0.8

0

Cp

(b)

ψ = 0ο

16ο

x/L = 0.650B1

−B2

−

−0.8

0

Cp

(c)

16ο

ψ = 0ο

x/L = 0.735B1

− B2

−

0 30 60 90 120 150 180Θ (deg.)

−0.8

0

Cp

(d)

ψ = 0ο

16ο

x/L = 0.840

B1

− B2

−

Figure 5. Circumferential distribution of Cp at selected x/L locations.The Cp contours for ψ = 0 ◦ are with no offset. For clarity, the suc-cessive contours for ψ ≥ 2 ◦ are vertically offset at incremental steps∆Cp =−0.05. The loci of minimum Cp are shown as dashed lines. Theseparation envelope (•−•) is modelled by Eqs. (1)-(3).

6(a)-(h). Here, the time scale t∗ increases with ψ, and as theflow separates the local maxima of Cp become more prominent.Figure 6(i) provides a further example which shows the effectof increasing ψ on selected contours “Cp =−0.10”. A plot ofthe size of the separation envelope ΘB−2

−ΘB−1from the con-

tours “Cp =−0.10” as a function of t∗ in figure 7 shows closeagreement with the power-law model (2), where the root-mean-square difference is ' 9%.

Concluding Remarks

Time-averaged surface-pressure maps have been obtained fromexperiments on inclined slender-body flows to allow compar-ison with separation lines established from flow visualisation.Each map is constructed from a grid of 33×360 ◦/4.5 ◦ = 2640pressure points. For body incidence angles up to 16 ◦, this ex-periment showed that the maximum difference between loadsobtained from integrating surface pressure and from internalstrain-gauge measurements is no more than 5%.

The most striking features of the surface flow are the negativebifurcations (B−1 and B−2 ). They propagate upstream with in-creasing body incidence angle, and they define the separationenvelope of the surface flow. Comparison with surface pressureshowed that the separation envelope is in the vicinity of the localpressure maxima (or plateaux) of the leeward flow.

B−1

B−2

B−1

B−2

B−1

B−2

B−1

B−2

B−1

B−2

B−1

B−2

B−1

B−2

B−1

B−2

B−1

B−2

*t t* t*

(a) ψ = 2o

(c) ψ = 6o

(b) ψ = 4o

(e) ψ = 10o

(f) ψ = 12o

(d) ψ = 8o

(g) ψ = 14o

(h) ψ = 16o

Cp = −0.10

o4

o2

o8

o6

o10

o16

o12 o

14

Cp Cp Cp

Cp Cp Cp

Cp Cp

Cp

Cp

Cp

0.450.400.350.300.250.200.150.100.05

0−0.05−0.10−0.15−0.20−0.25−0.30−0.35−0.40−0.45−0.50

0.50

0.450.400.350.300.250.200.150.100.05

0−0.05−0.10−0.15−0.20−0.25−0.30−0.35−0.40−0.45−0.50

0.50

0.450.400.350.300.250.200.150.100.05

0−0.05−0.10−0.15−0.20−0.25−0.30−0.35−0.40−0.45−0.50

0.50

o0

ψ

Θ(d

eg.)

180

120

60

0

Θ(d

eg.)

180

120

60

0

Θ(d

eg.)

180

120

60

0

(i)

0−0.05−0.10−0.15

0−0.05−0.10−0.15

0−0.05−0.10−0.15

0−0.05−0.10−0.15

0−0.05−0.10−0.15

0−0.05−0.10−0.15

0−0.05−0.10−0.15 −0.15

−0.10−0.05

0

100 2 6 84 100 2 6 84 100 2 6 84

Separationenvelope

with increasing ψ

Figure 6. (a)-(h) Surface pressure (Cp) as a function of time t∗; the bifurcations B−1 and B−2 are modelled by Eq. (2). (i) Example of selected contours“Cp =−0.10” showing the effect of increasing incidence angle ψ.

Acknowledgements

Thanks go to Mr. Paul Jacquemin and to QinetiQ for wind-tunnel operation and technical assistance, including the plumb-ing of pressure ports on the test geometry. This work acknowl-edges the financial support from DST Maritime Division.

References

[1] Henbest, S. M. and Jones, M. B., Transition on a genericsubmarine model. Defence Science and Technology, 2018,in preparation.

[2] Jeans, T. L. and Holloway, A. G. L., Flow-separation lineson axisymmetric bodies with tapered tails, J. Aircraft, 47(6),2010, 2177–2183.

[3] Lee, S.-K., Longitudinal development of flow-separationlines on slender bodies in translation, J. Fluid Mech., 837,2018, 627–639.

0 1 2 3 4 5 6t*

040

ΘB

2− −

ΘB

1−

(deg

.)

Contour Cp

= −0.10

Power-law model

10o

12o

16o

14o

()

ψ = 8ο

Figure 7. Comparison between the surface-flow separation envelope,ΘB−2−ΘB−1

, determined from the plateaux of selected contours “Cp =

−0.10” in figure 6(i) and from the power-law model (2). The error barsare defined by the resolution of the measurement grid for Cp.

[4] Wetzel, T. G., Simpson R. L. and Chesnakas C. J., Measure-ment of three-dimensional crossflow separation, AIAA J.,36(4), 1998, 557–564.