8.3.6. Steady Aerodynamics - 3-dimensional flows

todo

• Some remarks, using vorticity dynamics:

  • the starting vortex: when were vorticity and circulation generated?

  • vortex intensity of the wake: \(\boldsymbol\gamma_w(\mathbf{r}_w(\mathbf{r}_{TE})) = \Delta \boldsymbol\gamma(\mathbf{r}_{TE})\), from integral balance of vorticity in a stream-tube (todo check it and add details)

  • wake dynamics: transport of vorticity + vortex stretching; using vorticity equation, the shape of the wake \(\mathbf{r}_w(\mathbf{r}_{TE})\) and its intensity \(\boldsymbol\gamma(\mathbf{r}_w)\) can be computed (todo check it and add details) - or use Helmholtz’s vortex theorem + Kelvin’s circulation theorem (todo Add sections in the vorticity dynamics, or add a section here in Aerodynamics chapter, under the assumptions of almost-everywhere irrotational flow and negliglible viscosity effects.)

See \(\texttt{steady-aerodynamics-3d.md.bak}\) file for the old version of these notes.

8.3.6.1. Mathematical model

The governing equations of incompressible flows with negligible viscosity and almost everywhere irrotational can be recast as a Laplace equation, supplemented with the proper boundary conditions. In this model, the thin regions of the space where vorticity is non-negligible - namely boundary layers and wakes - need to be cut-out from the domain of the mathematical problem: boundary layers coincide with surfaces of solid bodies, while additional cuts must be introduced in the domain to represent wakes. These regions are represented by a (double-layer of) doublets or - by the doublet/vortex analogy - vortices. As shown in the section about singularities in incompressible aerodynamics todo, the potential is discontinuous across layers of doublets, the normal component of the velocity is continuous (for jump conditions), and the jump in tangential velocity can be related to the intensity of the vortex sheet, \(\Delta \mathbf{u} = - \hat{\mathbf{n}} \times \boldsymbol\gamma\).

Let the velocity field be \(\mathbf{u}(\mathbf{r}) = \nabla \phi(\mathbf{r})\) in the regions where the flow is irrotational, the governing equation and the no-penetration boundary conditions on surface of solid bodies at rest read

\[\begin{split}\begin{aligned} - \Delta \phi & = 0 && \mathbf{r} \in \Omega \\ \hat{\mathbf{n}} \cdot \boldsymbol\nabla \phi & = 0 && \mathbf{r} \in S_{body} \\ \end{aligned}\end{split}\]

In order to build a domain where the flow is irrotational, wake regions must be cut from the domain: both sides of these regions become part of the boundary of the domain. In an open domain, the boundary of the domain of the mathematical problem for the kinetic potential is the set union of body boundaries \(S_b\), wakes sides \(S_{w^+}\), \(S_{w^-}\), and a surface at the infinity \(S_{\infty}\),

\[\partial \Omega = S_b \cup S_{w^+} \cup S_{w^-} \cup S_{\infty} \ .\]

Boundary conditions at the wakes follow from jump conditions of mass and momentum across a wake, the balance of vorticity at trailing edges (Kutta condition) and the transport of vorticity, resulting from Helmholtz’s theorems. Boundary conditions at the boundary at the infinity \(S_{\infty}\) follow from the asymptotic behavior of the fields generated by singularities.

Usually the problem is written in terms of perturbation potential \(\varphi(\mathbf{r})\), separating the effects of bodies and wakes from the free-stream flow,

\[\begin{split}\begin{aligned} \mathbf{u}(\mathbf{r}) & = \mathbf{U}_{\infty} + \mathbf{u}'(\mathbf{r}) \\ \phi(\mathbf{r}) & = \mathbf{U}_{\infty} \cdot \mathbf{r} + \varphi(\mathbf{r}) \ , \end{aligned}\end{split}\]

so that Laplace equation and boundary conditions on \(S_{body}\) read

\[\begin{split}\begin{aligned} - \Delta \varphi & = 0 && \mathbf{r} \in \Omega \\ \hat{\mathbf{n}} \cdot \boldsymbol\nabla \varphi & = - \hat{\mathbf{n}} \cdot \mathbf{U}_{\infty} && \mathbf{r} \in S_{body} \\ \end{aligned}\end{split}\]

8.3.6.2. Green’s function method, for solving Poisson equations

Poisson equations can be recast as boundary problem and solved with a boundary element method, e.g. exploiting Green’s function properties. The Green’s function of a 3-dimensional Poisson problem reads

\[G(\mathbf{r}; \mathbf{r}_0) = \frac{1}{4\pi |\mathbf{r} - \mathbf{r}_0|} \ ,\]

so that the integral boundary value problems read

\[\begin{split}\begin{aligned} E(\mathbf{r}_0) \varphi(\mathbf{r}_0) & = - \oint_{\partial V} \left\{ \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \, \varphi(\mathbf{r}) - \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla \varphi(\mathbf{r}) \, G(\mathbf{r}; \mathbf{r}_0) \right\} \\ \end{aligned}\end{split}\]

The volume contribution of vorticity field is a function of the vorticity on the body todo see wake. Discuss and add details about this statement. For a given shape of the wake, the problem is linear: only the vorticity field, related to the stream function with linear dependence, is unkown; for a free-wake the problem becomes non-linear, as the shape of the wake - i.e. the position of its points - is unknown: this makes the problem non-linear, as the shape of the wake is determined by the condition of not-loaded wake.

\(E(\mathbf{r}_0)\)

The term \(E(\mathbf{r}_0)\) comes from the computation of the integral with the Dirac delta.

\[\begin{split}E(\mathbf{r}_0) = \begin{cases} 1 && \mathbf{r}_0 \in \Omega \\ \frac{1}{2} && \mathbf{r}_0 \in \partial \Omega \\ 0 && \mathbf{r}_0 \notin \Omega \cup \partial \Omega \end{cases}\end{split}\]

The value \(\frac{1}{2}\) is a particular case indeed, if the point \(\mathbf{r}_0\) is on a locally flat - no sharp angles - of the boundary. If the point is at a sharp angle - being \(\Theta\) the solid angle outside the domain - \(E(\mathbf{r}_0) = \frac{\Theta}{4 \pi}\).

Using the boundary conditions on \(S_b\), wake conditions on the \(S_{w^+}\) and \(S_{w^-}\) - collapsed in a single surface with normal pointing in the same direction of \(S_{w^+}\) - and asymptotic conditions on \(S_{\infty}\), the problem is recast as an integro-differential problem with unkown value of the potential \(\varphi(\mathbf{r})\) on1 the surface of the body \(S_b\),

\[\begin{split}\begin{aligned} & E(\mathbf{r}_0) \varphi(\mathbf{r}_0) + \int_{S_b} \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \varphi(\mathbf{r}) = \\ & = - \int_{S_b} \hat{\mathbf{n}}(\mathbf{r}) \cdot \mathbf{U}_\infty G(\mathbf{r}; \mathbf{r}_0) - \int_{S_w} \hat{\mathbf{n}} \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \Delta \varphi_w \ , \end{aligned}\end{split}\]

being the contributions on \(S_{\infty}\) equal to zero, and the jump of the potential across the wake a function of the jump at the trailing edge where the vortex line of the wake originates

\[\Delta \varphi(\mathbf{r}_w) = \Delta \varphi(\mathbf{r}_{TE}(\mathbf{r}_w)) \ .\]

Mathematical problem: prescribed wake. If the shape of the wake is known, the problem can thus be recast as a linear problem

\[\begin{aligned} E(\mathbf{r}_0) \varphi(\mathbf{r}_0) + \int_{\mathbf{r} \in S_b} \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \varphi(\mathbf{r}) + \int_{\mathbf{r}_w \in S_w} \hat{\mathbf{n}} \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \Delta \varphi(\mathbf{r}_{TE}(\mathbf{r}_w)) = - \int_{\mathbf{r} \in S_b} \hat{\mathbf{n}}(\mathbf{r}) \cdot \mathbf{U}_\infty G(\mathbf{r}; \mathbf{r}_0) \ , \end{aligned}\]

or in a discrete form

\[\left[ \mathbf{E} - \mathbf{D}_{bb} - \mathbf{D}_{bw} \mathbf{T}_{wb} \right] \boldsymbol\varphi_b = \mathbf{S}_{bb} \boldsymbol\sigma_b \ ,\]

being \(\mathbf{E}\) a multiple of the identity matrix, \(\mathbf{D}_{bb}\) the matrix containing the influence coefficients of body doublets on body collocation points, \(\mathbf{D}_{bw}\) the matrix containing the IC of the wake doubles on body collocation points, \(\mathbf{T}_{wb}\) the connectivity matrix relating the intensity of the wake to the jump of potential at the trailing edge, \(\mathbf{S}_{bb}\) the matrix containing the influence coefficients of body sources on body collocation points, with \(\left\{ \boldsymbol\sigma_b \right\}_i = - \hat{\mathbf{n}}_i \cdot \mathbf{U}_\infty\) the intensity of the body sources.

Mathematical problem: free wake. If the shape of the wake is unkown, the problem above must be supplemented with the conditions of not-loaded wake \(\mathbf{u}(\mathbf{r}_w) \times \boldsymbol\gamma(\mathbf{r}_w) = \mathbf{0}\). The problem becomes non-linear, so an iterative method is required.

Note. The problem is usually solved numerically by collocation. As the intensity of a vortex line - or a section of the vortex sheed representing the wake - is determined by a balance of vorticity at the trailing edge, only 2 components of the vector are effectively free. The transport of the wake occurs with the mean velocity \(\mathbf{u}(\mathbf{r}_w)\), and this velocity is aligned with the intensity of the vortex sheet \(\boldsymbol\gamma(\mathbf{r}_w) \parallel \mathbf{u}(\mathbf{r}_w)\). Being the free-stream velocity \(\mathbf{U}_\infty = U_\infty \hat{\mathbf{x}}\), the shape of the wake can be determined by the coordinates \((y_w, z_w)\) of the points of the wake on different planes with given values of \(x_w\).

Once the problem is solved and the potential \(\varphi(\mathbf{r})\) is known on solid surfaces \(\mathbf{r} \in S_b\) - and the shape of the wake, if unknown - it’s possible to retrieve the value of the potential and the velocity field in every point \(\mathbf{r}_0\) of the domain,

\[\begin{split}\begin{aligned} \varphi(\mathbf{r}_0) & = \int_{S_b} \sigma(\mathbf{r}) G(\mathbf{r}; \mathbf{r}_0) - \int_{S_b} \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \varphi(\mathbf{r}) - \int_{S_w} \hat{\mathbf{n}} \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \Delta \varphi_w \\ \mathbf{u}'(\mathbf{r}_0) & = \nabla_0 \varphi(\mathbf{r}_0) = \\ & = \nabla_0 \left\{ \int_{S_b} \sigma(\mathbf{r}) G(\mathbf{r}; \mathbf{r}_0) - \int_{S_b} \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \varphi(\mathbf{r}) - \int_{S_w} \hat{\mathbf{n}} \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \Delta \varphi_w \right\} = \\ & = \int_{S_b} \sigma(\mathbf{r}) \nabla_0 G(\mathbf{r}; \mathbf{r}_0) - \nabla_0 \int_{S_b} \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \varphi(\mathbf{r}) - \int_{S_w} \boldsymbol\gamma(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) \ , \end{aligned}\end{split}\]

having used the doublet/vortex analogy, where the line contributions are zero - \((a)\) contributions at trailing edge of the body and the wake are opposite, \((b)\) intensity of a wake goes to zero at its lateral boundaries, \((c)\) todo what’s the contribution of the starting vortex? is the starting vortex part of the domain? Discuss

Asymptotic behavior of the Green’s function - todo: move to section about singularities

For \(|\mathbf{r}_0| \gg |\mathbf{r}|\), Taylor expansion of the Green’s function \(G(\mathbf{r}; \mathbf{r}_0)\) and its gradient \(\boldsymbol\nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0)\) read

\[\begin{split}\begin{aligned} G(\mathbf{r}; \mathbf{r}_0) & \simeq G(\mathbf{r}; \mathbf{r}_0)|_{\mathbf{r} = \mathbf{0}} + \mathbf{r} \cdot \nabla_{\mathbf{r}} G(\mathbf{r}, \mathbf{r}_0)|_{\mathbf{r} = \mathbf{0}} = \\ & = \frac{1}{4 \pi}\frac{1}{|\mathbf{r}_0|} + \frac{1}{4 \pi} \mathbf{r} \cdot \frac{\mathbf{r}_0}{|\mathbf{r}_0|^3} \ . \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} \boldsymbol\nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0) & \simeq \boldsymbol\nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0)|_{\mathbf{r} = \mathbf{0}} + \mathbf{r} \cdot \nabla_{\mathbf{r}} \nabla_{\mathbf{r}} G(\mathbf{r}, \mathbf{r}_0)|_{\mathbf{r} = \mathbf{0}} = \\ & = \frac{1}{4 \pi} \frac{\mathbf{r}_0}{|\mathbf{r}_0|^3} + \frac{1}{4\pi} \left[ - \dfrac{\mathbf{r}}{|\mathbf{r}_0|^3} + 3 \mathbf{r} \cdot \mathbf{r}_0 \frac{\mathbf{r}_0}{|\mathbf{r}_0|^5} \right] = \\ & = \frac{1}{4 \pi} \frac{\mathbf{r}_0 - \mathbf{r}}{|\mathbf{r}_0|^3} + \frac{3}{4 \pi} \mathbf{r} \cdot \mathbf{r}_0 \frac{\mathbf{r}_0}{|\mathbf{r}_0|^5} \ . \end{aligned}\end{split}\]

First and second space derivatives of the Green’s function read

\[\begin{split}\begin{aligned} \nabla_{\mathbf{r}} |\mathbf{r}_0 - \mathbf{r}|^{-1} & = \mathbf{\hat{x}}_i \dfrac{\partial}{\partial x_i} | \mathbf{r}_0 - \mathbf{r} |^{-1} = \\ & = - \mathbf{\hat{x}}_i \frac{x_{i} - x_{0,i}}{| \mathbf{r}_0 - \mathbf{r} |^3} = \\ & = - \frac{\mathbf{r} - \mathbf{r}_0}{| \mathbf{r}_0 - \mathbf{r} |^3} \ . \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} - \nabla_{\mathbf{r}} \frac{\mathbf{r} - \mathbf{r}_0}{| \mathbf{r}_0 - \mathbf{r} |^3} & = - \mathbf{\hat{x}}_k \dfrac{\partial}{\partial x_k} \left( \mathbf{\hat{x}}_i \frac{x_{i} - x_{0,i}}{| \mathbf{r}_0 - \mathbf{r} |^3} \right) = \\ & = - \mathbf{\hat{x}}_k \mathbf{\hat{x}}_i \left[ \dfrac{\delta_{ij} |\mathbf{r}_0 - \mathbf{r}|^3 - 3 (x_i - x_{0,i})(x_k - x_{0,k})| \mathbf{r}_0 - \mathbf{r} | }{| \mathbf{r}_0 - \mathbf{r} |^6} \right] = \\ & = - \frac{1}{| \mathbf{r}_0 - \mathbf{r} |^3} \, \mathbb{I} + 3 \frac{(\mathbf{r}-\mathbf{r}_{0}) \otimes (\mathbf{r}-\mathbf{r}_{0})}{| \mathbf{r}_0 - \mathbf{r} |^5} \ . \end{aligned}\end{split}\]

Doublet/vortex equivalence

A continuous distribution of doublets over a surface \(S\) is equivalent to the sum of a surface distribution of vortices over \(S\) and a line distribution of vortices over \(\partial S\). The perturbation potential in \(\mathbf{r}_0 \notin S\) generated by a surface distribution of doublets with intensity \(\mu(\mathbf{r})\) on \(S\) reads

\[\varphi(\mathbf{r}_0) = \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) \ .\]

Exploiting the properties, \(\nabla G = - \nabla_0 G\), the potential can be recast as

\[\varphi(\mathbf{r}_0) = - \nabla_0 \cdot \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) \ ,\]

so that the velocity reads

\[\begin{split}\begin{aligned} \mathbf{u}(\mathbf{r}_0) & = \nabla_0 \varphi(\mathbf{r}_0) = \\ & = - \nabla_0 \left( \nabla_0 \cdot \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) \right) = \\ & = - \nabla_0^2 \left( \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) \right) - \nabla_0 \times \nabla_0 \times \left( \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) \right) = \\ & = - \nabla_0 \times \nabla_0 \times \left( \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) \right) = \\ & = - \nabla_0 \times \left( \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) \right) = \\ & = \nabla_0 \times \boldsymbol\psi(\mathbf{r}_0) \ , \end{aligned}\end{split}\]

having introduced the definition of vector potential - or stream function - \(\boldsymbol\psi(\mathbf{r}_0)\), and having used the vector identity, \(\nabla^2 \mathbf{v} = \nabla ( \nabla \cdot \mathbf{v} ) - \nabla \times \nabla \times \mathbf{v}\), and the definition of the Green’s function for Poisson equation to see the first integral is identically zero,

\[\begin{split}\begin{aligned} \nabla_0^2 \left( \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) \right) & = \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) \nabla_0^2 G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) = \\ & = - \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) \delta(\mathbf{r}-\mathbf{r}_0) \, \mu(\mathbf{r}) = \\ & = \mathbf{0} \ , \quad \text{for $\mathbf{r}_0 \notin S$} \ , \end{aligned}\end{split}\]

and having exploited the relations \(\nabla \times ( \mathbf{a} f(\mathbf{r}) ) = \mathbf{a} \times \nabla f(\mathbf{r})\), \(\nabla^2 (\mathbf{a} f(\mathbf{r}) = \mathbf{a} \nabla^2 f(\mathbf{r})\), for constant \(\mathbf{a}\).

The expression of the vector potential can be rearranged using integration by parts (be careful about the domain of the functions, and the differential operators. See below),

\[\begin{split}\begin{aligned} \boldsymbol\psi(\mathbf{r}_0) & = - \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) = \\ & = \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) \times \nabla G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) = \\ & = \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) \times \nabla \left( G(\mathbf{r}; \mathbf{r}_0) \, \mu(\mathbf{r}) \right) - \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) \times \nabla^s \mu(\mathbf{r} ) \, G(\mathbf{r}; \mathbf{r}_0) = \\ & = \oint_{\mathbf{r}\in \partial S} \hat{\mathbf{t}}(\mathbf{r}) \, \mu(\mathbf{r}) \, G(\mathbf{r}; \mathbf{r}_0) - \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) \times \nabla^s \mu(\mathbf{r} ) \, G(\mathbf{r}; \mathbf{r}_0) = \\ \end{aligned}\end{split}\]

so that a doublet distribution of intensity \(\mu\) over surface \(S\) is equivalent to the sum of a vortex distribution of intensity \(-\hat{\mathbf{n}}(\mathbf{r}) \times \nabla^s \mu(\mathbf{r})\) over surface \(S\) and a vortex distribution of intensity \(\mu(\mathbf{r}) \hat{\mathbf{t}}(\mathbf{r})\) over the boundary \(\partial S\) of the surface, being \(\nabla^s \mu(\mathbf{r})\) the surface gradient, defined for function on a surface with coordinates \(q^i\), \(i=1:2\),

\[\nabla^s f = \mathbf{b}^i \frac{\partial f}{\partial q^i} \ ,\]

that can be defined for functions defined not only on the surface as the projection of the gradient on the surface,

\[\nabla^s = \left( \mathbb{I} - \hat{\mathbf{n}} \otimes \hat{\mathbf{n}} \right) \cdot \nabla \ ,\]

so that \(\nabla^s f = \nabla f - \hat{\mathbf{n}} \hat{\mathbf{n}} \cdot \nabla f \ .\)

In the manipulation of the expression of the vector potential, a vector identity, see the second of the two useful lemmas in vector calculus is used to transform the first integral, in Cartesian components

\[\oint_{\partial S} A t_l = \int_S n_i \varepsilon_{ijk} \partial_j ( A \delta_{kl} ) = \int_S n_i \varepsilon_{ijl} \partial_j A \]

or in vector formalism

\[\oint_{\partial S} A \hat{\mathbf{t}} \cdot \hat{\mathbf{e}}_l = \int_S \hat{\mathbf{n}} \cdot \nabla ( A \hat{\mathbf{e}}_l ) = \int_S \hat{\mathbf{n}} \times \nabla A \cdot \hat{\mathbf{e}}_l \ , \]

or, without projecting on the constant unit vector \(\hat{\mathbf{e}}_l\) the first and the last expressions,

\[\oint_{\partial S} A \hat{\mathbf{t}} = \int_S \hat{\mathbf{n}} \times \nabla A \ .\]

The perturbation velocity is retrieved from the vector potential as \(\mathbf{u}(\mathbf{r}_0) = \nabla_0 \times \boldsymbol\psi(\mathbf{r}_0)\),

\[\begin{split}\begin{aligned} \mathbf{u}'(\mathbf{r}_0) & = \nabla_0 \times \boldsymbol\psi(\mathbf{r}_0) = \\ & = \nabla_0 \times \left[ \oint_{\mathbf{r}\in \partial S} \mu(\mathbf{r}) \hat{\mathbf{t}}(\mathbf{r}) \, G(\mathbf{r}; \mathbf{r}_0) - \int_{\mathbf{r}\in S} \hat{\mathbf{n}}(\mathbf{r}) \times \nabla^s \mu(\mathbf{r} ) \, G(\mathbf{r}; \mathbf{r}_0) = \\ \right] = \\ & = \oint_{\mathbf{r}\in \partial S} \mu(\mathbf{r}) \hat{\mathbf{t}}(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) - \int_{\mathbf{r}\in S} \left[ \hat{\mathbf{n}}(\mathbf{r}) \times \nabla^s \mu(\mathbf{r} ) \right] \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) \ . \end{aligned}\end{split}\]

Vorticity field from the velocity field - (todo? Trash?)

Given the velocity field

\[\mathbf{u}(\mathbf{r}_0) = \nabla_0 \times \boldsymbol\psi(\mathbf{r}_0) + \nabla_0 \varphi(\mathbf{r}_0) \ ,\]

the vorticity field reads

\[\begin{split}\begin{aligned} \boldsymbol\omega(\mathbf{r}_0) & = \nabla_0 \times \mathbf{u}(\mathbf{r}_0) = \\ & = \nabla_0 \times \left[ \nabla_0 \times \boldsymbol\psi(\mathbf{r}_0) + \nabla_0 \varphi(\mathbf{r}_0) \right] = \\ & = \nabla_0 \times \left( \nabla_0 \times \boldsymbol\psi(\mathbf{r}_0) \right) = \\ & = \nabla_0 \left( \nabla_0 \cdot \boldsymbol\psi(\mathbf{r}_0) \right) - \nabla^2_0 \boldsymbol\psi(\mathbf{r}_0) \ , \end{aligned}\end{split}\]

with the vector identity \(\nabla^2 \mathbf{v} = \nabla (\nabla \cdot \mathbf{v}) - \nabla \times \nabla \times \mathbf{v}\). With uniform free-stream the vorticity can be evaluated using perturbation velocity,

\[\begin{split}\begin{aligned} \boldsymbol\omega(\mathbf{r}_0) & = \nabla_0 \times \mathbf{u}_{\gamma}(\mathbf{r}_0) = \\ & = \nabla_0 \times \left[ \oint_{\mathbf{r}\in \partial S_{b,w,\infty}} \mu(\mathbf{r}) \hat{\mathbf{t}}(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) - \int_{\mathbf{r}\in S_{b,w,\infty}} \left[ \hat{\mathbf{n}}(\mathbf{r}) \times \nabla^s \mu(\mathbf{r} ) \right] \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) \right] = \\ & = \nabla_0 \times \left[ \oint_{\mathbf{r}\in \partial S_{b,w,\infty}} \boldsymbol\Gamma(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) + \int_{\mathbf{r}\in S_{b,w,\infty}} \boldsymbol\gamma(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) \right] = \\ \end{aligned}\end{split}\]

todo Provide the meaning of \(\boldsymbol\Gamma\), \(\boldsymbol\gamma\) as (impulsive) vorticity distribution on lines and surfaces, using

• identity

  \[\begin{split}\begin{aligned} \nabla \times ( \mathbf{a} \times \nabla f(\mathbf{r}) ) & = \hat{\mathbf{x}}_i \varepsilon_{ijk} \partial_j \varepsilon_{klm} a_l \partial_m f = \\ & = \hat{\mathbf{x}}_i \left( \delta_{il} \delta_{jm} - \delta_{im} \delta_{jl} \right) a_l \partial_{jm} f = \\ & = \hat{\mathbf{x}}_i \left( a_i \partial_{ll} f - a_l \partial_{il} f \right) = \\ & = \mathbf{a} \nabla^2 f(\mathbf{r}) - ( \mathbf{a} \cdot \nabla ) \nabla f(\mathbf{r}) = \\ & = \mathbf{a} \nabla^2 f(\mathbf{r}) - \nabla ( \mathbf{a} \cdot \nabla f(\mathbf{r}) ) \ , \end{aligned}\end{split}\]

  for \(\mathbf{a}\) constant.

• \(- \nabla_0^2 G(\mathbf{r}; \mathbf{r}_0) = \delta(\mathbf{r} - \mathbf{r}_0)\)

• …

  \[\begin{split}\begin{aligned} \nabla_0 \cdot \left( \nabla_0 G(\mathbf{r}; \mathbf{r}_0) \right) & = \frac{\partial}{\partial x^0_i} \left[ \frac{x_i - x_i^0)}{|\mathbf{r}-\mathbf{r}_0|^3} \right] = \\ & = \frac{-3}{|\mathbf{r}-\mathbf{r}_0|^3} + 3 \frac{( x_i - x^0_i)(x_i - x^0_i)}{|\mathbf{r}-\mathbf{r}_0|^5} = 0 \ , \end{aligned}\end{split}\]

  when \(\mathbf{r} \ne \mathbf{r}_0\).

• …

  \[\begin{split}\begin{aligned} 4 \pi \nabla_0 \left( \mathbf{a} \cdot \nabla_0 G(\mathbf{r}; \mathbf{r}_0) \right) & = \nabla_0 \left( \mathbf{a} \cdot \frac{\mathbf{r}-\mathbf{r}_0}{|\mathbf{r}-\mathbf{r}_0|^3} \right) = \\ & = \hat{\mathbf{x}}^0_i \frac{\partial}{\partial x^0_i} \left[ \frac{a_k(x_k - x_k^0)}{|\mathbf{r}-\mathbf{r}_0|^3} \right] = \\ & = \hat{\mathbf{x}}^0_i \left[ \frac{ - a_k \delta_{ik} }{|\mathbf{r}-\mathbf{r}_0|^3} + 3 \frac{\mathbf{a}\cdot(\mathbf{r}-\mathbf{r}_0)( x_i - x^0_i)}{|\mathbf{r}-\mathbf{r}_0|^5} \right] = \\ \end{aligned}\end{split}\]

\(\oint_{\mathbf{r}_0 \in \ell_0} \hat{\mathbf{t}}_0 \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0)\) - (todo? Trash?)

…

Asymptotic behavior of the perturbation potential

Body. For \(|\mathbf{r}_0| \gg |\mathbf{r}|\), using Taylor expansion of the Green’s function \(G(\mathbf{r}; \mathbf{r}_0)\) and its gradient \(\boldsymbol\nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0)\), the contribution of the body to perturbation velocity reads

\[\begin{split}\begin{aligned} \mathbf{u}'_b(\mathbf{r}_0) & = \int_{\mathbf{r}\in S_b} \boldsymbol\gamma(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) + \int_{\mathbf{r} \in S_b} \sigma(\mathbf{r}) \nabla_0 G(\mathbf{r}; \mathbf{r}_0) = \\ & \simeq \int_{\mathbf{r}\in S_b} \boldsymbol\gamma(\mathbf{r}) \times \left[ \frac{1}{4 \pi} \frac{\mathbf{r}_0}{|\mathbf{r}_0|^3} - \frac{1}{4 \pi} \frac{\mathbf{r}}{|\mathbf{r}_0|^3} + \frac{3}{4\pi} \mathbf{r} \cdot \mathbf{r}_0 \frac{\mathbf{r}_0}{|\mathbf{r}_0|^5} \right] + \\ & \quad + \int_{\mathbf{r} \in S_b} \sigma(\mathbf{r}) \left[ \frac{1}{4 \pi} \frac{\mathbf{r}_0}{|\mathbf{r}_0|^3} - \frac{1}{4 \pi} \frac{\mathbf{r}}{|\mathbf{r}_0|^3} + \frac{3}{4\pi} \mathbf{r} \cdot \mathbf{r}_0 \frac{\mathbf{r}_0}{|\mathbf{r}_0|^5} \right] \ . \end{aligned}\end{split}\]

The first term in the second integral is proportional to the mass flux \(\dot{m} = \int_{\mathbf{r} \in S_b} \hat{\mathbf{n}} \cdot \nabla_{\mathbf{r}} \varphi = \int_{\mathbf{r} \in S_b} \hat{\mathbf{n}} \cdot \mathbf{u}\) across \(S_b\), and so it’s identically zero if \(\dot{m} = 0\), as in the case of solid boundaries. The asymptotic behavior of the perturbation velocity due to the body as \(|\mathbf{r}_0| \gg |\mathbf{r}|\) reads

\[\mathbf{u}'_b(\mathbf{r}_0) \simeq \frac{1}{4 \pi} \boldsymbol\Gamma \times \frac{\mathbf{r}_0}{|\mathbf{r}_0|^3} + \frac{1}{4 \pi} \dot{m} \frac{\mathbf{r}_0}{|\mathbf{r}_0|^3} + O\left( \frac{1}{|\mathbf{r}_0|^3} \right) \]

Wake.

8.3.6.3. Wake and the shape of the domain

Jump conditions. Jump conditions across the wake read

• for mass

  \[\left[ \rho u_n^{rel} \right] = 0 \ .\]

• for momentum, with negligible viscosity so that \(\mathbf{t}_{\mathbf{n}} = - P \hat{\mathbf{n}}\)

  \[\left[ \rho \mathbf{u} u_n^{rel} + P \hat{\mathbf{n}} \right] = \mathbf{0} \ .\]

As the wake is (todo Prove or justify) a contact discontinuity, with no mass flux through it, \(\dot{m} = \rho u_n^{rel} = 0\) (a stricter condition w.r.t. to no jump in mass flux across the surface), the momentum jump condition across the wake reads

\[[ P \hat{\mathbf{n}} ] = 0 \ ,\]

i.e. the vector condition contains a non-trivial 1-dimensional information, along the normal direction only \(\left[ P \right] = 0\).

Using second Bernoulli theorem, connecting a pair of points on the two sides of the wake (if needed, do an intermediate step connecting each one of these points with a point at infinity) gives

\[P^+ + \frac{1}{2} \rho \left| \mathbf{u}^+ \right|^2 = P^- + \frac{1}{2} \rho \left| \mathbf{u}^- \right|^2 \ .\]

Using the relation between the velocity on each side of the wake, the average velocity, the velocity jump and the intensity of vortex sheet of the wake,

\[\begin{split}\begin{aligned} \mathbf{u}^+ & = \mathbf{u} + \frac{\Delta \mathbf{u}}{2} \\ \mathbf{u}^- & = \mathbf{u} - \frac{\Delta \mathbf{u}}{2} \\ \Delta \mathbf{u} & = - \hat{\mathbf{n}} \times \boldsymbol\gamma \ , \end{aligned}\end{split}\]

gives

\[\begin{split}\begin{aligned} 0 = \left[ P \right] = \\ & = P^+ - P^- = \\ & = \frac{1}{2} \rho \left[ |\mathbf{u}|^2 + \frac{|\Delta \mathbf{u}|^2}{4} + \mathbf{u} \cdot \Delta \mathbf{u} - |\mathbf{u}|^2 - \frac{|\Delta \mathbf{u}|^2}{4} + \mathbf{u} \cdot \Delta \mathbf{u} \right] = \\ & = \mathbf{u} \cdot \Delta \mathbf{u} = \\ & = - \mathbf{u} \cdot \hat{\mathbf{n}} \times \boldsymbol\gamma = \\ & = \hat{\mathbf{n}} \cdot \mathbf{u} \times \boldsymbol\gamma \ . \end{aligned}\end{split}\]

As (todo discuss) \(\mathbf{u}\) and \(\boldsymbol\gamma\) are everywhere orthogonal to \(\hat{\mathbf{n}}\), and thus tangent to the wake surface, it follows that

\[\mathbf{u} \times \boldsymbol\gamma = \mathbf{0} \ ,\]

or, equivalently,

\[\begin{split} \boldsymbol\gamma \parallel \mathbf{u} \qquad \text{or} \qquad \begin{aligned} \boldsymbol\gamma & = \gamma \, \hat{\mathbf{t}} \\ \mathbf{u} & = u \, \hat{\mathbf{t}} \ . \end{aligned} \end{split}\]

Physical conditions: jump conditions, transport

…

8.3.6.4. Aerodynamic forces

Lift and drag

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What does the value of the potential on a surface doublet mean, when it’s not continuous there? Different equivalent approaches can be used. It’s possible to take a point \((a)\) «just above» the surface, s.t. \(E(\mathbf{r}_0) = 1\) and the singular part of the doublet integral reads \(\frac{1}{2} \varphi(\mathbf{r}_0)\), \((b)\) «just below» the surface s.t. \(E(\mathbf{r}_0) = 0\) and the singular part of the doublet integral reads \(-\frac{1}{2} \varphi(\mathbf{r}_0)\), or \((c)\) «on» the surface s.t. \(E(\mathbf{r}_0) = \frac{1}{2}\) and the singular part of the doublet integral reads \(0\) by regularization.