EFFECTS OF CLOSURE MODELS ON SUPERCAVITATING BUBBLE DEVELOPMENT

Journal of Marine Science and Technology, Vol. 13, No. 3, pp. 163-169 (2005) 163

EFFECTS OF CLOSURE MODELS ONSUPERCAVITATING BUBBLE DEVELOPMENT

Bin-Chen Wu* and Jiahn-Horng Chen*

Paper Submitted 06/30/04, Accepted 03/16/05. Author for Correspondence:Jiahn-Horng Chen. E-mail: b0105@mail.ntou.edu.tw.*Department of Systems Engineering and Naval Architecture National

Taiwan Ocean University Keelung, Taiwan, R.O.C.

Key words: 2-D supercavitation, closure model, hydrofoil.

ABSTRACT

Effects of different cavity closure models on the developmentsheet supercavitation of two-dimensional hydrofoils were examinedin this report. The study was carried out under the assumption ofpotential flow for which a cavity closure model was needed. Apotential-based boundary element method has been developed for thispurpose. The models we employed included the constant velocitymodel and the short plate termination model. Several different airfoilshapes were tested. Their effects on the cavity shape, lift coefficient,cavity volume and computation efficiency were studied. Carefulinvestigations show that closure models have insignificant influenceon physical quantities, but do have strong effects on computationalefficiency.

INTRODUCTION

Cavitation is a challenging hydrodynamic phe-nomenon both in physical science and engineeringapplications. In physical science, cavitating flows in-volve many complicated and interdisciplinary physicalphenomena in which both physicists and applied math-ematicians are interested. In engineering applications,erosion and corrosion of structures are almost unavoid-able wherever cavitation occurs. Detrimental noiseensues from collapse of cavitating bubbles. Morerecently, new generation of underwater weapons hasbeen developed by the concept of supercavitation.

In the past two decades, there has been significantprogress in the study and observation of sheet cavitationphenomena, primarily due to the fast development ofhigh-speed computation science and more advancedexperimental techniques. Nevertheless, the investiga-tion of cavitation modeling has a very long history thatcan be traced back to Kirchhoff [8]. Major pioneeringdevelopment was theoretical [5, 9]. The successful

application of potential flow theory played an activerole in these early theoretical developments. Literaturesurveys indicate that there are several significant fea-tures worth of attention. The cavity was usually macro-scopically treated as a single big bubble of finite length,which encompasses the whole region where cavitationsoccur and micro bubbles dominate. In addition, thepressure inside the cavity bubble was usually assumedto be constant. Linearized theories were proposed byTulin [13, 14]. He obtained the analytical solution fora supercavitating flat plate hydrofoil with a sharp lead-ing edge. Later, Acosta [1] provided the first partialcavitation solution for a flat-plate hydrofoil. Morerecent linearized theoretical development was due toFuruya and Acosta [6]. Nonlinear development hasbeen explored by Wu [18, 19]. He employed the open-wake model to study the cavitating flat plate at anarbitrary angle of flow incidence.

However, in the context of potential flow, thespecification of condition in the cavity closure regioncreates serious problems in cavitation modeling. It isclear that very complicated flow phenomena occur inthis region and cannot be incorporated into a potentialflow model. In fact, the termination of cavity has not yetbeen clearly described due to its turbulent, two-phase,and even compressible flow physics. Therefore, it isnecessary to resort to some artifact in the vicinity in thisregion to cope with this difficulty. Several closuremodels have been devised (see, e.g. [2]). The modelsinvestigated in the present study will be addressed anddiscussed later.

With the modern evolution of computationalmethods, several nonlinear numerical procedures ofboundary-element type have been successfully devel-oped for the solution of sheet cavitation, based on theearly theoretical achievements. In these approaches,cavity surface conditions are usually satisfied on theexact cavity surface that is part of the solution anddetermined iteratively by proper computationalalgorithms. Uhlman [15, 16] employed a velocity-based nonlinear boundary element method to obtainsolutions for partially-cavitating and supercavitating

Journal of Marine Science and Technology, Vol. 13, No. 3 (2005)164

hydrofoil flows. Kinnas and Fine [7] developed apotential-based nonlinear boundary element method.This appears to be superior to the velocity-based meth-ods in terms of convergence. Several different cavityclosure models have been incorporated in these compu-tation procedures. For example, in partially cavitatingflow computations, Kinnas and Fine [7] employed amodel in which the velocity in the termination zone wasdecreased to zero according to a prescribed law. Morerecently, the reentrant jet was used by Krishnaswamy etal. [10]. Vaz et al. [17] developed a new boundaryelement method in which the elements are located on thefoil surface and the boundary conditions for the cavitysurface have been reformulated based upon a Taylorexpansion. They compared their results with thoseusing the reentrant jet model [4] and concluded thatdifferent models had insignificant effects on the cavityphysics.

In supercavitating flow computations, Uhlman [16]employed the short-plate model; Lee et al. [11] andChen and Weng [3] used a simple closure model.However, their effects on computations, such as con-vergence and convergence rate, and on flow physics,such as lift, cavity shape, and so on, have not yet beenstudied and compared in detail. It is the purpose of thepresent study to carry out a more clear observation incomputation.

THEORETICAL FORMULATION

Consider a uniform potential flow past a two-dimensional hydrofoil, as shown in Figure 1. Thegoverning equation of the velocity potential φ = φ(x) isthe Laplace equation,

∇2φ = 0. (1)

The flow velocity is related to the velocity poten-tial by

u = ∇φ. (2)

Several boundary conditions are identified for well-posedness of the problem.

For a uniform incoming flow, the velocity distri-bution far away from the hydrofoil is

∇φ = U (icosα + jsinα), (3)

where α is the angle of attack of the uniform flow, U thespeed of the incoming uniform flow, and i and j the unitvectors in x- and y-directions, respectively.

On the surface of hydrofoil, Sb, and the surface ofcavity, Sc, a kinematic condition must be prescribed,

∇φ • n = 0, (4)

where n denotes the unit normal to the surfaces. Thiscondition guarantees no penetration of the fluid flowthrough the surfaces.

In addition, a dynamic condition is required on thesurface of the cavity, Sc. That is, the pressure inside thecavity is constant,

p = pv = constant, (5)

where pv denotes the vapor pressure. According to thiscondition, it can be derived via the Bernoulli equationthat on the cavity surface, we have

|uc| = constant, (6)

where uc represents the tangential velocity on the cavitysurface. The implication of this condition is that thevelocity potential at any point x on the cavity surfacecan be expressed as

φ(x) = φ(xd) + |uc|s, (7)

where the subscript d represents the detachment point ofthe cavity and s the distance between the detachmentpoint and x along the cavity surface.

Furthermore, the cavitation number, σ, can berelated to uc as follows,

σ =P – p v

12ρU 2

=uc

U

2

– 1, (8)

where P is the pressure of the undisturbed flow and thedensity of fluid.

For a cavitating flow, we need two more conditions,i.e., the detachment condition and the termination con-dition of the cavity. These two conditions prescribewhere the cavity begins and how it ends. Generallyspeaking, these conditions cannot be discussed pre-cisely within the context of potential flow since the flowdetachment points are strongly affected by viscous ef-fect and the flow around the cavity end is obviouslyturbulent. For simplicity, we assume that the flowdetaches exactly at the leading edge. It is also assumed

Foil Sbα Cavity Sc

P = PV

Fig. 1. Schematic of cavitating flow.

B.C. Wu & J.H. Chen: Effects of Closure Models on Supercavitating Bubble Development 165

that the flow detaches at the trailing edge for asupercavitating flow.

The termination condition and the related modelswill be discussed in detailed in the next section.

Finally, for a two-dimensional potential flow in amulti-connected domain, the Kutta condition must beprescribed to determine uniquely the circulation. Thiscondition is prescribed at the end point of cavity for asupercavitating flow.

CAVITY CLOSURE MODELS

Two different closure models were studied. Webriefly describe and comment on these models in thefollowing.

The first one is the simple closure model which,shown in Figure 2, prescribes a simple close cavity onwhich surface the tangential velocity is constanteverywhere, according to Eqs. (5) and (6). This modelwas employed in [11]. The cavity shape and the tangen-tial speed on the cavity surface are part of the solutionto be determined.

This is a purely computational model due to itssimplicity in numerical implementation. In fact, fromthe mathematical point of view, a closed finite bodywith a constant pressure does not exist in exact potentialflow theory. This kind of termination condition iscontradictory to the constant pressure condition insidethe cavity (or the constant tangential velocity conditionon the cavity surface).

The second model is the short plate terminationmodel (or modified Riabouchinsky model). This is awell-developed model proposed by Riabouchinsky [12].As shown in Figure 3, the cavity ends at a plate normalto the incoming flow. In addition to the cavity shape andthe tangential velocity, the height of the short plate isunknown and need be determined as the flow solution issought.

The short plate model was first employed byUhlman [16]. Incorporating with this model, he devel-oped a velocity-based nonlinear boundary elementscheme to compute supercavitating flow fields.

NUMERICAL DISCRETIZATION

According to Green’s identity, the solution of the

Laplace equation, Eq. (1) can be written in terms ofnormal dipoles and sources distributed on the boundarysurfaces. Here, we introduced a dipole distribution onthe cavity and wet foil surfaces and a source distributionon the cavity surface. The total potential of the flowmay be expressed as

φ(x ) = U ⋅ x + qS c

(ξ)φsdS – µS b + S c

(ξ)(n ⋅ ∇ φx) dS

+ µwS w

(ξ) (n ⋅ ∇ φx) dS

where q(ξ) and µ(ξ) represent the source and dipolestrengths at position ξ, respectively, φx is the potentialdue to a source of unit strength located at x on theboundary surface, µw the dipole strength on the wakesurface Sw, and n the unit normal directed into the flowat ξ.

The distribution of sources on the cavity surfaceserves as a normal flux generator so as to adjust the formof cavity surface. They vanish pointwise when thecavity surface is exactly prescribed or located, leavingonly a dipole distribution on the foil and cavity surfaces.

The numerical discretization has been a standardprocedure. The wetted part of the hydrofoil surface andthe cavity surface are approximated by an N-sided poly-gon defined by Nb and Nc vertex points (“nodes”) dis-tributed on the surfaces Sb and Sc, respectively. Thetotal number of elements N is the sum of Nb and Nc.Assume that the dipole and source strengths on eachelement are constant and that the control point is posi-tioned at the center of each element. The discretizedform the integral formulation can be formally written as

φi = U ⋅ xi + Σj = 1

N c

q jαij – Σk = 1

N

µkβ ik + µwβ iw (9)

where the subscript i denotes the control point of i-thelement and the coupling coefficient is defined asfollows,

αij = 12π

lnS j

rdS

End point

Foil Cavity

Wake

Fig. 2. The constant-pressure model.

Cavity surface

Wake

Foil Short plate = ± /2θ π

Inflow

Fig. 3. The modified riabouchinsky model.

Journal of Marine Science and Technology, Vol. 13, No. 3 (2005)166

β ik =

12π S j

∂ln r∂n k

dS , i ≠ k ,

12

, i = k (10)

where r is the distance between points ξ and x.To obtain the solution, one must solve together the

unknown cavity shape and discretized source and dipolestrengths in the equation system (9). Therefore, thesingularity strengths and locations on the cavity surfacemust be updated iteratively. A nonlinear procedure hasbeen developed to cope with this kind of solution. Thedetailed description of the iterative procedure can befound in [3]. For completeness, it is briefly stated in thefollowing.(1) Prescribe the cavity length and an initial shape of

cavity surface.(2) Determine the dipole and source strengths by solv-

ing the equation system of Eq. (9) on the basis of theprescribed cavity boundary.

(3) The shape of cavity surface is then updated based onthe source distribution obtained in the last step.

(4) Repeat steps (2) and (3) till reasonable convergenceis achieve.

SOME TEST CASES AND DISCUSSIONS

For test purposes, a series of computations havebeen conducted for three different hydrofoil sections.They are flat plate, NACA16004 section, and Eppler’snew section. The flat-plate hydrofoil is the most typicalcase in literature because its analytical data are available.The NACA16004 section represents a “traditional” hy-drofoil section. In contrast, the Eppler’s new sectionrepresents a hydrofoil section of new generation de-signed specifically for cavitating flow conditions. Foreach hydrofoil, we varied the cavity-length-to-chordratio ( / c = 1.2, 1.4, 1.6, 1.8, 2.0) and the angle ofattack (α = 2°, 4°, 6°, 8°).

In addition to compare effects of different closuremodels on the flow development, we employed the samemesh in the chordwise direction for the each flow setupcondition. For the short plate model, two additionalelements were implemented on the plate to model plateeffects. Some important effects we have observed arediscussed in the following.

We first examine the cavity shape. Some of typicalconvergent computational results for the simple andshort-plate models are shown in Figures 4-7. Figures 4and 5 show the convergence history (L2-norm of differ-ence between two subsequent iterative solutions) andthe cavity shape, respectively, for different hydrofoilsections at a small flow angle of attack, α = 4°, and ashort cavity length, / c . On the other hand, Figures 6and 7 show results at a bigger angle of attack, α = 8°, anda longer cavity length / c .

For each case, regardless of the incidence angles

1.00E+01

1.00E-01

1.00E-03

1.00E-05

1.00E-07

1.00E-09

1.00E-110 50 100 150

Numbers of iterations

(a) Flat plate.

Simple model

Plate model

L2-

norm

200 250 300

1.00E+011.00E-011.00E-031.00E-051.00E-071.00E-091.00E-11

0 50 100 150Numbers of iterations

(b) NACA16004

L2-

norm

200 250 300

1.00E+011.00E-011.00E-031.00E-051.00E-071.00E-091.00E-11

0 50 100 150Numbers of iterations

(c) Eppler plate.

L2-

norm

200 250

Fig. 4. Convergence history for different hydrofoils at α = 4° and / c = 1.2.

0.300.250.200.150.100.050.00

-0.05

Plate model

Simple model

y

x/c

0.00 0.20 0.40(a) Flat plate.

1.000.800.60 1.20 1.40

0.120.100.080.060.040.020.00

-0.02-0.04

y

x/c

0.00 0.20 0.40(b) NACA16004

0.60 0.80 1.401.00 1.20

0.15

0.10

0.05

0.00

-0.10

y

x/c

0.00 0.20 0.40(c) Eppler foil.

0.60 1.201.000.80 1.40

Fig. 5. Cavity shapes for different hydrofoils at α = 4° and / c = 1.2.

B.C. Wu & J.H. Chen: Effects of Closure Models on Supercavitating Bubble Development 167

of flow and cavity lengths, we find that the cavity shapecurves closely coincide to each other, except at the rearcavity region where the closure models are applied.This seems to indicates that different closure models donot significantly alter the cavity shape.

It is interesting to note that, as pointed out in theprevious section, the simple closure model contradictsto the potential flow theory and, therefore, cannot be areal physical model. However, this fact does not createa significant deviation of cavity shape. Therefore, as faras computations and engineering applications areconcerned, this seems to be a feasible model. In fact, wefind that the simple closure model is a good one in termsof not only its engineering applicability but also itsrobust convergence property that will be discussed later.

In addition, since the cavity shapes computed fromdifferent closure models do not differ significantly, wemay conclude immediately that the cavity volumes varyinsignificantly.

The cavitation numbers we obtained by computa-tions with three different closure models do not varysignificantly. Furthermore, general observationsthroughout our test cases also show that the static pres-sure distributions on the foil surfaces agree with oneanother very well. This implies that at a specified set offlow incidence and cavity length, the lifts due to differ-ent closure models do not have observable deviationfrom one another.

Observing these computational results, we mayconclude that, as far as macroscopic properties of cavityare concerned, cavity closure models have only minoreffects on the development of flows of two-dimensionalsupercavitating hydrofoils.

Nevertheless, the convergence history of itera-tions shows an interesting phenomenon for differentclosure models. As shown in Figures 4 and 6, it appearsthat computations with the simple closure model have amuch better and stable convergence history. In contrast,those employing the short-plate model usually requiremuch more number of iterations to converge. What isworse is that the iterations sometimes lead to divergence.

Figure 8 shows the convergence history at a = 2°and / c = 1.4. Iterated with the same initial guess, thecomputations that incorporated with the simple modelwere fully convergent within 100 iterations whereasthose with the short-plate model could not achieve areasonable convergence. In fact, we found that for allthree hydrofoils the computations with the short-platemodel were usually divergent for a long cavity if theangle of attack was small. It seems that the short-platemodel is somewhat unstable for iterations of the presentpotential-based scheme. Nevertheless, it appears thatthis model does not show such divergent iterations inthe velocity-based scheme developed by Uhlman [16] inwhich the unknown is the surface vorticity rather thanvelocity potential. The reason why the employment ofthe modified Riabouchinsky model in the present poten-tial-based boundary element method leads to numericalinstability in some cases is not clear. A possible cause

1.00E+01

1.00E-01

1.00E-03

1.00E-05

1.00E-07

1.00E-09

1.00E-110 100 200 300

Numbers of iterations

(a) Flat plate.

Simple model

Plate model

L2-

norm

400 500 600 700

1.00E+01

1.00E-01

1.00E-03

1.00E-05

1.00E-07

1.00E-09

1.00E-110 200 400

Numbers of iterations

(b) NACA16004

L2-

norm

600 800 1000

1.00E+01

1.00E-01

1.00E-03

1.00E-05

1.00E-07

1.00E-09

1.00E-110 100 200 300

Numbers of iterations

(c) Eppler foil.

L2-

norm

400 500

Fig. 6. Convergence history for different hydrofoils at α = 8° and/ c = 2.0.

0.300.250.200.150.100.050.00

-0.05

Plate model

Simple modely

x/c

0.00 0.50 1.00(a) Flat plate.

1.50 2.00 2.50

0.30

0.20

0.10

0.00

-0.10

y

x/c

0.00 0.50 1.00(b) NACA16004

1.50 2.00 2.50

0.40

0.30

0.20

0.10

0.00

-0.10

y

x/c

0.00 0.50 1.00(c) Eppler foil.

1.50 2.00 2.50

Fig. 7. Cavity shapes for different hydrofoils at α = 8° and / c = 2.0.

Journal of Marine Science and Technology, Vol. 13, No. 3 (2005)168

1.0E+03

1.0E+01

1.0E-01

1.0E-03

1.0E-05

1.0E-07

1.0E-09

1.0E-11

Simpe model (N = 200)

0 10 20 30

Numbers of iterations

L2

norm

40 50 60

Plate model (N = 202)

Fig. 8. Convergence history for flat-plate hydrofoils at α = 2° and / c = 1.4.

may be due to sharp edges of streamline near the ends ofthe short plate, which is detrimental in finding thetangential velocity on the cavity surface by taking firstderivatives of the velocity potential. This is especiallytrue for a long bubble with a small angle of attack. Thevelocity-based scheme avoids this dilemma because itemploys surface vorticity as unknown from which thevelocity distribution can be immediately obtained with-out taking partial derivatives.

Nevertheless, as the angle of attack increases, theflat-plate model leads to convergence. Figure 9 showssuch an example for which the angle of attack is and thecavity length ratio is 1.4. However, it is evident that the

computation with the flat-plate model took much longertime to converge, compared to that with the simpleclosure model. The latter converged within 100iterations, whereas the former did not fully convergebefore 1000 iterations.

For an even larger angle of attack, the convergencehistory became better for the flat-plate model, as shownin Figure 10. Nevertheless, the simple closure model isstill a better scheme to be employed in computationsbecause of its better convergence rate. Similar behav-iors of convergence were observed for other hydrofoils.It seems that these trends are universal.

CONCLUSION

Our study shows that cavity closure models do nothave strong effect on the development of the cavity andengineering integrated properties of interests insupercavitating flows. Even though the reentrant jetmodel is known to its better approximation to realsupercavitating flows, the computation results showthat they do not deviate significantly from those bysimpler closure models.

What should be more cautious in using these mod-els is the convergence problem in computations. Dif-ferent models incur different convergence processesand in some cases result in divergence during iterations.Therefore, as far as engineering applications areconcerned, the simple closure model should be a betterchoice because of its easiest implementation and nu-merical robustness in iterations.

ACKNOWLEDGEMENTS

This work was supported by the National ScienceCouncil, Republic of China, under the grant NSC91-2815-C-019-013-E. The authors would like to expresstheir thanks to this support.

REFERENCES

1. Acosta, A.J., A Note on Partial Cavitation of Flat PlateHydrofoils (Hydrodynamics Laboratory Report E19.9),California Institute of Technology, Pasadena, CA (1955).

2. Brennen, C.E., Cavitation and Bubble Dynamics, Ox-ford University Press, NewYork (1955).

3. Chen, J.-H. and Weng, Y.-C., “Analysis of Flow Past aTwo-Dimensional Supercavitating Hydro-Foil Using aSimple Closure Model,” P. Natl. Sci. Counc., ROC(A),Vol. 23, pp. 411-418 (1999).

4. Dang, J. and Kuiper, G., “Re-entrant Jet Modeling ofPartial Cavity Flow on Two-Dimensional Hydrofoils,”Proceedings of the Third Symposium on Cavitation,Grenoble, France (1998).

1.0E+01

1.0E-01

1.0E-03

1.0E-05

1.0E-07

1.0E-09

1.0E-11

Simpe model (N = 200)

0 200 400 600

Numbers of iterations

L2

norm

800 1000 12000

Plate model (N = 202)

1.0E+01

1.0E-01

1.0E-03

1.0E-05

1.0E-07

1.0E-09

1.0E-11

Simpe model (N = 200)

0 50 100 150

Numbers of iterations

L2

norm

200 250 300 350

Plate model (N = 202)

Fig. 10. Convergence history for NACA16004 hydrofoils at α = 6° and/ c = 1.4.

Fig. 9. Convergence history for NACA16004 hydrofoils at α = 4° and/ c = 1.4.

B.C. Wu & J.H. Chen: Effects of Closure Models on Supercavitating Bubble Development 169

5. Efros, D.A., “Hydrodynamical Theory of Two-Dimen-sional Flow with Cavitation,” Dokl. Akad. Nauk SSSR,Vol. 51, pp. 267-270 (1946).

6. Furuya, O. and Acosta, A.J., “A Note on the Calculationof Supercavitating Hydrofoils with Rounded Noses,” J.Fluid Eng-T ASME, Vol. 95, pp. 222-228 (1973).

7. Kinnas, S.A. and Fine, N.E., “A Numerical NonlinearAnalysis of the Flow Around Two and Three-Dimen-sional Partially Cavitating Hydrofoils,” J. Fluid Mech.,Vol. 254, pp. 151-181 (1993).

8. Kirchhoff, G., “Zur Theorie freier Flussigkeitsstrahl-en,” Z. Reine Angew. Math., Vol. 70, pp. 289-298 (1869).

9. Kreisel, G., Cavitation with Finite CavitationNumbers (Technical Report No. R1/H/36), AdmiraltyResearch Laboratory, Teddington, UK (1946).

10. Krishnaswamy, P., Andersen, P., and Kinnas, S.A., “Re-entrant Jet Modeling for Partially Cavitating Two-Di-mensional Hydrofoils,” Proceedings of the 4th Interna-tional Symposium on Cavitation, CAV2001, Pasadena,CA (2001).

11. Lee, C.S., Kim, Y.G., and Lee, J.T., “A Potential-BasedPanel Method for the Analysis of a Two-DimensionalSuper- or Partially-Cavitating Hydrofoil,” J. Ship Res.,Vol. 36, pp. 168-181 (1992).

12. Riabouchinsky, D., “On Steady Fluid Motion with a Free

Surface,” P. Lond. Math. Soc., Vol. 19, pp. 206-215(1920).

13. Tulin, M.P., Steady Two-Dimensional Cavity FlowsAbout Slender Bodies (Report No. 834), David TaylorModel Basin, Washington, DC (1953).

14. Tulin, M.P., “Supercavitating Flows - Small Perturba-tion Theory,” J. Ship Res., Vol. 7, pp. 16-37 (1964).

15. Uhlman, J.S., “The Surface Singularity Method Appliedto Partially Cavitating Hydrofoils,” J. Ship Res., Vol. 31,pp. 107-124 (1987).

16. Uhlman, J.S., “The Surface Singularity or BoundaryIntegral Method Applied to Supercavitating Hydro-foils,” J. Ship Res., Vol. 33, pp. 16-20 (1989).

17. Vaz, G., Bosschers, J., and Falcão de Campos, J.A.C.,“Analysis of Two Boundary Element Methods for Cavi-tation Modeling,” Proceeding of the 5th InternationalSymposium on Cavitation, CAV2003, Osaka, Japan(2003).

18. Wu, T.Y., “A Free Streamline Theory for Two-Dimen-sional Fully Cavitated Hydrofoils,” J. Math. Phys., Vol.35, pp. 236-265 (1956).

19. Wu, T.Y., “A Wake Model for Free Streamline FlowTheory, Part 1. Fully and Partially Developed WakeFlows and Cavity Flows past an Oblique Flat Plate,” J.Fluid Mech., Vol. 13, pp. 161-181 (1962).