7 Theoretical
7.11 Source and vortex panel methods
Classical hydrodynamic theory shows that flow about a body can be generated by using the appropriate distribu- tions of sources, sinks, vortices and dipoles distributed both within and about itself. Increased computational power has led to the development of panel methods, and these have now become commonplace for the solution of potential flow problems about arbitrary bodies.
In the case where the body generates no lift the flow field can be computed by replacing the surface of the body with an appropriate distribution of source panels (Figure 7.31). These source panels effectively form a source sheet whose strength varies over the body sur- face in such a way that the velocity normal to the body surface just balances the normal component of the free stream velocity. This condition ensures that no flow passes through the body and its surface becomes a streamline of the flow field. For practical computation
Theoretical methods – basic concepts 165
Figure 7.31 Source panel solution method
purposes, the source strengthλjis assumed to be con- stant over the length of thejth panel but allowed to vary from one panel to another. The mid-point of the panel is taken as the control point at which the resultant flow is required to be a tangent to the panel surfaces, thereby satisfying the flow normality condition defined above. The end points of each panel, termed the bound- ary points, are coincident with those of the neighbouring panels and consequently form a continuous surface.
From an analysis of this configuration ofnpanels as shown in Figure 7.31, the total velocity at the surface at theith control point, located at the mid-point of theith panel, is given by the sum of the contributions of free stream velocityV and those of the source panels:
Vi=V∞sinδ+ n j=1
λj
2π
j
∂
∂s(lnrij)dsj (7.57)
Figure 7.32 Vortex panel solution method
from which the associated pressure coefficient is given by
CPi=1− Vi
V∞ 2
(7.57a) In equation (7.57)ris the distance from any point on thejth panel to the mid-point on theith panel andsis the distance around the source sheet.
When the body for analysis is classified as a lifting body, the alternative concept of a vortex panel must be used since the source panel does not possess the cir- culation property which is essential to the concept to lift generation. The modelling procedure for the vortex panel is analogous to that for the source panel in that the body is replaced by a finite number of vortex panels as seen in Figure 7.32. On each of the panels the cir- culation density is varied from one boundary point to
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166 Marine propellers and propulsion
the other and is continuous over the boundary point. In these techniques the Kutta condition is easily introduced and is generally stable unless large numbers of panels are chosen on an aerofoil with a cusped trailing edge.
As in the case of the source panels the boundary points and control points are located on the surface of the body; again with control points at the mid-panel position. At these control points the velocity normal to the body is specified so as to prevent flow through the aerofoil. Using this approach the velocity potential at theith control point (xi,yi) is given by
φ(xi,yi)=V∞(xicosα+yisinα)
− m
j=1
γ(sj) 2π tan−1
yi−yj xi−xj
dsj
(7.58) for a system ofmvortex panels, with
γ(sj)=γj+(γj+1−γj)sj Sj
(7.58a) Methods of this type – and the outline discussed here is but one example – can be used in place of the clas- sical methods, for example Theodorsen or Weber, to calculate the flow around aerofoils. Typically for such a calculation one might use fifty or so panels to obtain the required accuracy, which presents a fairly extensive numerical task as compared to the classical approach.
However, the absolute number of panels used for a particular application is dependent upon the section thickness to chord ratio in order to preserve the stability of the numerical solution. Nevertheless, methods of this type do have considerable advantages when consider- ing cascades or aerofoils with flaps, for which exact methods are not available.
In a similar manner to the classical two-dimensional methods, source and vortex panel methods can be extended to three-dimensional problems. However, for the panel methods the computations become rather more complex but no new concepts are involved.
References and further reading
1. Milne-Thomson, L.M.Theoretical Aerodynamics.
Dover, New York, 1973.
2. Milne-Thomson, L.M.Theoretical Hydrodynam- ics. Macmillan, 1968.
3. Lighthill, J.An Informal Introduction to Theoretical Fluid Mechanics. Oxford, 1986.
4. Anderson, J.D. Fundamentals of Aerodynamics.
McGraw-Hill, 1985.
5. Prandtl, L., Tietjens, O.G.Applied Hydro- and Aeromechanics. United Engineering Trustees, 1934; also Dover, 1957.
6. Burrill, L.C. Calculation of marine propeller per- formance characteristics.Trans NECIES,60, 1944.
7. Hawdon, L., Carlton, J.S., Leathard, F.I. The analy- sis of controllable pitch propellers characteristics at off-design conditions.Trans. I.Mar.E,88, 1976.
8. Pankhurst, R.C. A Method for the Rapid Evaluation of Glauert’s Expressions for the Angle of Zero Lift and the Moment at Zero Lift. R. & M. No. 1914, March 1944.
9. Theodorsen, T. Theory of Wing Sections of Arbi- trary Shape. NACA Report No. 411, 1931.
10. Theodorsen, T., Garrick, I.E. General Potential The- ory of Arbitrary Wing Sections. NACA Report No.
452, 1933.
11. Abbott, H., Doenhoff, A.E. van.Theory of Wing Sections. Dover, 1959.
12. Pinkerton, R.M. Calculated and Measured Pres- sure Distributions over the Mid-Span Sections of the NACA 4412 Airfoil. NACA Report No. 563, 1936.
13. Riegels, F., Wittich, H.Zur Bereschnung der Druck- verteilung von Profiles. Jahrbuch der Deutschen Luftfahrtforschung, 1942.
14. Weber, J. The Calculation of the Pressure Distri- bution on the Surface of Thick Cambered Wings and the Design of Wings with Given Pressure Distributions. R. & M. No. 3026, June 1955, HMSO.
15. Wilkinson, D.H. A Numerical Solution of the Analysis and Design Problems for the Flow Past one or more Aerofoils in Cascades. R. & M. No.
3545, 1968, HMSO.
16. Firmin, M.C.P. Calculation of the Pressure Distri- bution, Lift and Drag on Aerofoils at Subcritical Conditions. RAE Technical Report No. 72235, 1972.
17. Schlichting, H.Boundary Layer Theory. McGraw- Hill, 1979.
18. Owen, R.R., Klanfer, L. RAE Report Aero, 2508, 1953.
19. Oossanen, P. van.Calculation of Performance and Cavitation Characteristics of Propellers Including Effects of Non-Uniform Flow and Viscosity. NSMB Publication 457, 1970.
20. Curle, N., Skau, S.W. Approximation methods for predictory separation properties of laminar boundary layers.Aeronaut. Quart.,3, 1957.
21. Smith, A.M.O. Transition, pressure gradient, and stability theory. Proc. 9th Int. Congress of App.
Mech., Brussels,4, 1956.
22. Nash, J.F., Macdonald, A.G.J. The Calculation of Momentum Thickness in a Turbulent Boundary Layer at Mach Numbers up to Unity. ARC Paper C.P. No. 963, 1967.
23. Jones, W.P., Launder, B.E. The Prediction of Laminarization with a Two-Equation Model of Turbulence.Int. J. Heat Mass Trans.,15, 1972.