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According to Katz and Plotkin [34], the potential lifting surface problem can be divided into three subproblems, with its respective potentials Φ1, Φ2 and Φ3, as follows:

1. Straight mean-line foil shape, as shown in Figure 3.3 with internal finite volume, at zero incidence angle, with the main equation ∇2_Φ

Figure 3.3: Scheme of the perturbation potential Φ and its subproblems Φ1, Φ2 and Φ3

condition given by equation (3.10)

∂Φ1 ∂y =±

∂ηt

∂xQ∞ (3.10)

whereηtaccounts for the normal surface coordinate of uncambered non-zero thick-

ness foil with signs + for the upper surface ηt(u) and −for the lower surface ηt(l);

2. Zero-thickness, uncambered foil mean-line at an incidence angle with main equa- tion ∇2_Φ

2 = 0 and the boundary condition given by equation (3.11); ∂Φ2

∂y =−Q∞α (3.11)

3. Zero-thickness cambered foil mean-line at zero incidence angle, with main equa- tion ∇2_Φ

3 = 0 and the boundary condition given by equation (3.12) ∂Φ3

∂y = ∂ηc

∂xQ∞ (3.12)

where ηc accounts for the normal surface coordinate of the cambered thin foil.

The solution of the potential subproblem 1 can be done by the distribution of source singularities in two ways: By distributing sources on body surface, as shown in Figure 3.4 for surface distribution, where the source strengths are given by the difference of the internal and external potentials or; by distributing sources along the meanline of

body, as shown in Figure 3.4, and the source strengths are given by the difference in

ηt between upper and lower potentials of surfaces.

Figure 3.4: Scheme of possible configurations for source distributions

Subproblems 2 and 3 can be solved together, becoming one non-symmetric subprob- lem: zero-thickness cambered airfoil at an incidence angle. Two kinds of singularities can be used to model the problem: doublet and vortex. A more common approach considers that singularities are related to the potential jump between the upper and lower sides of the thin foil. However, the distribution of doublets or vortex on a thick body surface is possible and will be discussed later in this section.

According to Katz and Plotkin [34], the perturbation potential is derived following Green’s identity and it is given in equation (3.13) for any point on body surface SB,

where σ is the source and µthe dipole strengths. Φ = −1 4π Z SB σ 1 r −µn· ∇ 1 r ds+ Φ∞ (3.13)

In the equation (3.13), inside the brackets, the subproblem 1 corresponds to the left hand side, while subproblems 2 and 3 correspond to the right hand side.

3.1.4.1 The Application of Tangential Boundary Condition

The basic approaches to apply tangential boundary condition on body surface are Neumman and Dirichlet.

The Neumman Boundary Condition The Neumman boundary condition is simply

the consideration of zero normal velocity on body surface ∂Φ/∂n = 0. It implies on evaluating the resulting velocity field generated by the contribution of potential subproblems, meaning the entire solution of equation (3.13). Katz and Plotkin [34] call it as direct boundary condition. It is given by equation (3.4).

The Dirichlet Boundary Condition The Dirichlet boundary condition considers that at a distance r far from body, the flow disturbance is zero.

lim

r→∞∇Φ = 0

If the condition ∂Φ/∂n = 0 on body surface is required then, potential inside body (without internal singularities) will not change (Φ0 =constant). This constant can be

specified as zero. Applying Neumann condition, as discussed by Katz and Plotkin [34], solution will be equivalent to equation (3.14).

∂Φ ∂n =−n·Q∞ (3.14) Considering −σ = ∂Φ ∂n − ∂Φ0 ∂n

and Φ0 = 0, with a constant potential function then, ∂_∂Φ_n0 = 0 too. the Dirichlet

boundary condition will be given in equation (3.15), where σ is a source strength and

n points inside the body.

σ=n·Q∞ (3.15)

Dirichlet boundary condition is applied to source singularities, implying that body needs a finite volume inside so, the constant potential can be adopted. By establishing the constant potential, the condition of tangential flow on surface is satisfied indirectly so, it is called by Katz and Plotkin [34] as indirect method.

The Dirichlet boundary condition, for viscous-inviscid interaction, as it will be seen in Chapter 5, allows a split solution for inviscid and viscous flows and the viscous part can be inserted as a separate module in calculation.

3.1.4.2 Surface and Lattice Methods

The source singularity modelling, discussed before, can be done by two approaches: Using a distribution of sources on a meanline inside surface or, by distributing them on wing surface. The same approaches can be used for doublet or vortex singularities. The difference between them is the tangential flow boundary condition used. The distribution of singularities on a body surface (surface panel method) uses normally the Dirichlet boundary condition, as an internal potential has to be specified.

The problem of singularities distributed on a meanline does not need a specification of an internal potential but, the boundary condition has to be satisfied at the body surface points then, Neumman condition is applied. The combination of potentials of all singularities involved has to satisfy the tangential flow condition. The meanline distribution of singularities is also known as lattice methods, according to Couser [28]. One advantage of lattice methods over the surface panel method is the number of panels involved in calculation as described in works of Greeley and Cross-Whiter [52]

and Couser [53]. Lattice methods (LM) use fewer panels but, according to Hess [40], LM has less accuracy than surface method due to small oscillations between lattices. Hess [40] obtained 20% of error comparing the LM and his own surface panel method.

3.1.4.3 The Kutta Condition

According to Hess and Smith [54], describing flow over thick and non-lifting bodies with source singularities distribution is sufficient. However, for the lifting cases, a boundary condition has to be specified at the trailing edge, in order to have for resulting body circulation Γ an unique solution and a finite value for velocity at this region.

The Kutta boundary condition states that: “The flow leaves the sharp trailing edge of an airfoil smoothly and the velocity there is finite”. This is interpreted by Katz and Plotkin [34] as a trailing edge flow with the following characteristics:

• Flow leaving sharp trailing edge along the bisector line there;

• The normal component of trailing edge velocity must vanish for both sides of foil (upper and lower);

• The pressure jump at trailing edge is zero;

∆pT E = 0

• If circulation is modelled by a vortex distribution, pressure jump can be expressed as

γT E = 0

where T E subscripts mean trailing edge region.

3.1.4.4 The Wake Modelling

Doublets or vortices are used to solve the subproblems 2 and 3 discussed before: zero thickness cambered foil at an angle of attack. The most important variable for subprob- lems 2 and 3 is the amount of circulation Γ generated by the body. In two-dimensional flow, a trailing vortex segment of wake is not necessary since it has zero vorticity and it is sufficient to specify the location of trailing edge where Kutta condition has to be satisfied.

In three-dimensional flow, according to Katz and Plotkin [34], if wing is looked at a distance, it can be modelled as a vorticity segment producing a determinate circulation. According to the Helmholtz theorem [34], vorticity segments cannot begin and end in the fluid. After reaching the surface limits, vorticity vector turns to be parallel to the local velocity vector and they shed at some length into flow, as shown in Figure 3.5.

If foil is discretized into many spanwise lines of vorticity, these lines will come out of wing in different points along the trailing edge and form the wake. As wake cannot

Figure 3.5: Scheme of vortex segment development in a finite wing according to Helmholtz theorem

generate force in the fluid, the lines coming out of trailing edge should be parallel to the local flow direction at any point and, as observed by Katz and Plotkin [34], the vortex singularity strength γW along this line or, the doublet strength µW, must be

constant.

According to Hess [40] and Katz and Plotkin [34], the wake modelling is part of the solution and, the disturbed potential equation (3.13) should take into account the wake vorticity. Then, equation (3.13) becomes (3.16) for three-dimensional flow.

Φ = 1 4π Z body+wake µn· ∇ 1 r dS− 1 4π Z body σ 1 r dS+ Φ∞ (3.16)