8.3.5. Steady Aerodynamics - 2-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 \(\gamma(\mathbf{r}_w)\) 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.)

8.3.5.1. Mathematical model

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

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. As shown below, there’s no physical wake for 2-dimensional steady flows, while wakes exist in 3-dimensional flows.

8.3.5.2. Green’s function method

8.3.5.2.1. 2-dimensional problems

Green’s function. Green’s function of a 2-dimensional Poisson problem reads

\[G(\mathbf{r}; \mathbf{r}_0) = - \frac{1}{2\pi} \ln |\mathbf{r} - \mathbf{r}_0| \ .\]

Note! When taking the logarithm of a physical quantity, that quantity must be non-dimensional. Here, the problem must be made non-dimensional, as an example scaling all the lengths with a reference length.

Derivation of the Green’s function for 2-dimensional Poisson problem.

Solutions with spherical symmetry of the Poisson equation in 2-dimensional domains, in the whole domain except for the origin of the coordinates (or the position of the active point, \(\mathbf{r}_0\), as the problem is independent from translation of coordinates under the assumption of homogeneous domain)

\[-\Delta G = \delta(r) \ .\]

The problem is regular for \(r \ne 0\)

\[-\Delta G \quad , \quad |\mathbf{r}| \ne 0 \ ,\]

and it can be solved using the expression of the Laplacian for spherical symmetric scalar fields

\[0 = \frac{1}{r} \left( r G'(r) \right)' \ ,\]

and integrating twice

\[\begin{split}\begin{aligned} G'(r) & = \frac{A}{r} \\ G (r) & = A \ln r + B \\ \end{aligned}\end{split}\]

The additive constant \(B\) is set to zero, while the constant \(A\) is evalauted by the condition

\[1 = \int_{\Omega} \delta(r) = - \int_{\Omega} \Delta G = - \oint_{\partial \Omega} \hat{\mathbf{n}} \cdot \boldsymbol\nabla G = - \oint_{C_r} \hat{\mathbf{r}} \cdot \hat{\mathbf{r}} \partial_r G = - \oint_{C_r} \frac{A}{r} = - \int_{\theta=0}^{2\pi} \frac{A}{r} \, r d\theta = - 2 \pi A \ ,\]

so that \(A = -\frac{1}{2 \pi}\), and

\[G(\mathbf{r}; \mathbf{r}_0) = - \frac{1}{2\pi} \ln |\mathbf{r} - \mathbf{r}_0| \ .\]

Solution.

\[\begin{split}\begin{aligned} E(\mathbf{r}_0) \phi(\mathbf{r}_0) & = \int_{\mathbf{r}\in\Omega} \delta(\mathbf{r}-\mathbf{r}_0) \phi(\mathbf{r}) = \\ & = - \int_{\mathbf{r}\in\Omega} \nabla^2_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0) \phi(\mathbf{r}) = \\ & = - \int_{\mathbf{r}\in\Omega} \nabla_\mathbf{r} \cdot \left( \nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0) \phi(\mathbf{r}) - G(\mathbf{r}; \mathbf{r}_0) \nabla_{\mathbf{r}} \phi(\mathbf{r}) \right) - \int_{\mathbf{r}\in\Omega} G(\mathbf{r};\mathbf{r}_0) \underbrace{\nabla^2_{\mathbf{r}} \phi(\mathbf{r})}_{=0 \text{ , Poisson eq}} = \\ & = - \int_{\mathbf{r}\in \partial \Omega} \left\{ \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0) \phi(\mathbf{r}) - \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla_{\mathbf{r}} \phi(\mathbf{r}) G(\mathbf{r}; \mathbf{r}_0) \right\} \ . \end{aligned}\end{split}\]

As shown below, in 2-dimensional steady problems there’s no physical wake, i.e. \(\partial \Omega = S_{body} \cup S_{\infty}\), or equivalently on \(S^+_w\)and \(S^-_w\),

\[0 = \hat{\mathbf{n}}^- \cdot \mathbf{u}^- + \hat{\mathbf{n}}^+ \cdot \mathbf{u}^+ = \ ...\]

…

For the perturbation potential, \(\varphi := \phi - \phi_\infty = \phi - \mathbf{U}_\infty \cdot \mathbf{r}\), with \(\varphi|_{S_\infty} = 0\) (todo prove it! And find the trend \(\propto r^{\alpha}\))

\[\begin{split}\begin{aligned} E(\mathbf{r}_0) \varphi(\mathbf{r}_0) & = - \int_{\mathbf{r}\in \partial \Omega} \left\{ \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0) \varphi(\mathbf{r}) - \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla_{\mathbf{r}} \varphi(\mathbf{r}) G(\mathbf{r}; \mathbf{r}_0) \right\} = \\ & = - \int_{\mathbf{r}\in S_b} \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0) \varphi(\mathbf{r}) + \int_{\mathbf{r} \in S_b} \underbrace{ \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla \varphi(\mathbf{r})}_{= - \hat{\mathbf{n}} \cdot \mathbf{U}_\infty \text{ , from b.c.}} G(\mathbf{r}; \mathbf{r}_0) + \\ & \quad - \int_{\mathbf{r}\in S_w} \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0) \, \underbrace{ \Delta \varphi(\mathbf{r})}_{ = \Delta \varphi_{TE} \text{ , from wake c.} } \ . \end{aligned}\end{split}\]

This equation can be recast as an integro-differential equation for the perturbation potential on the surface of the body \(\mathbf{r}_0 \in S_{b}\) that can be solved by collocation, as an example using boundary element methods. Once this problem is solved, the perturbation potential on \(S_b\) is known, and the perturbation potential can be computed in every point of the domain.

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

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 angle outside the domain - \(E(\mathbf{r}_0) = \frac{\theta}{2 \pi}\).

Asymptotic behavior of the perturbation velocity

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

\[\begin{split}\begin{aligned} G(\mathbf{r}; \mathbf{r}_0) & \simeq G(\mathbf{r}; \mathbf{r}_0)|_{\mathbf{r} = \mathbf{0}} + \mathbf{r} \cdot \nabla_{\mathbf{r}} G(\mathbf{r}, \mathbf{r}_0)|_{\mathbf{r} = \mathbf{0}} = \\ & = - \frac{1}{2 \pi} \ln |\mathbf{r}_0| + \mathbf{r} \cdot \frac{1}{2 \pi} \frac{\mathbf{r}_0}{|\mathbf{r}_0|^2} \ . \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} \boldsymbol\nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0) & \simeq \boldsymbol\nabla_{\mathbf{r}} G(\mathbf{r}; \mathbf{r}_0)|_{\mathbf{r} = \mathbf{0}} + \mathbf{r} \cdot \nabla_{\mathbf{r}} \nabla_{\mathbf{r}} G(\mathbf{r}, \mathbf{r}_0)|_{\mathbf{r} = \mathbf{0}} = \\ & = \frac{1}{2 \pi} \frac{\mathbf{r}_0}{|\mathbf{r}_0|^2} + \frac{1}{2\pi} \left[ - \dfrac{\mathbf{r}}{|\mathbf{r}_0|^2} + 2 \mathbf{r} \cdot \mathbf{r}_0 \frac{\mathbf{r}_0}{|\mathbf{r}_0|^4} \right] = \\ & = \frac{1}{2 \pi} \frac{\mathbf{r}_0 - \mathbf{r}}{|\mathbf{r}_0|^2} + \frac{1}{\pi} \mathbf{r} \cdot \mathbf{r}_0 \frac{\mathbf{r}_0}{|\mathbf{r}_0|^4} \ . \end{aligned}\end{split}\]

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

\[\begin{split}\begin{aligned} - \nabla_{\mathbf{r}} \ln |\mathbf{r}_0 - \mathbf{r}| & = - \mathbf{\hat{x}}_i \dfrac{\partial}{\partial x_i} \ln | \mathbf{r}_0 - \mathbf{r} | = \\ & = - \mathbf{\hat{x}}_i \frac{x_{i} - x_{0,i}}{| \mathbf{r}_0 - \mathbf{r} |^2} = \\ & = - \frac{\mathbf{r} - \mathbf{r}_0}{| \mathbf{r}_0 - \mathbf{r} |^2} \ . \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} - \nabla_{\mathbf{r}} \frac{\mathbf{r} - \mathbf{r}_0}{| \mathbf{r}_0 - \mathbf{r} |^2} & = - \mathbf{\hat{x}}_k \dfrac{\partial}{\partial x_k} \left( \mathbf{\hat{x}}_i \frac{x_{i} - x_{0,i}}{| \mathbf{r}_0 - \mathbf{r} |^2} \right) = \\ & = - \mathbf{\hat{x}}_k \mathbf{\hat{x}}_i \left[ \dfrac{\delta_{ij} |\mathbf{r}_0 - \mathbf{r}|^2 - 2 (x_i - x_{0,i})(x_k - x_{0,k}) }{| \mathbf{r}_0 - \mathbf{r} |^4} \right] = \\ & = - \frac{1}{| \mathbf{r}_0 - \mathbf{r} |^2} \, \mathbb{I} + 2 \frac{(\mathbf{r}-\mathbf{r}_{0}) \otimes (\mathbf{r}-\mathbf{r}_{0})}{| \mathbf{r}_0 - \mathbf{r} |^4} \ . \end{aligned}\end{split}\]

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 \(\mathbf{u}(\mathbf{r}_0) = \nabla_0 \times \boldsymbol\psi(\mathbf{r}_0)\),

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

Zero wake vorticity in steady 2-dimensional problems

As the potential jump across the wake in steady 2-dimensional problem is uniform along the wake, \(\mu^w = \Delta \varphi^w\), then \(\nabla^s \mu^w = \mathbf{0}\) and

\[\boldsymbol\gamma^w = \mathbf{0} \ .\]

Details in integration by parts on surface in doublet/vortex equivalence

todo Using surface coordinates, a surface nabla seamlessly appears

Relationship between doublet/vortex distribution and the circulation - 2-dimensional steady problems

In 2-dimensional steady problems there’s no physical wake - no velocity discontinuities, zero vorticity intensity, \(\boldsymbol\gamma(\mathbf{r}^w)= \mathbf{0}\). For closed surfaces \(|\partial S_b| = 0\), and thus the velocity field reads (with the unit normal vector pointing into the fluid domain, reversed w.r.t. the original expression at the beginning of this section),

\[\begin{aligned} \mathbf{u}'(\mathbf{r}_0) & = \int_{\mathbf{r} \in S_b} \boldsymbol\gamma(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) + \int_{\mathbf{r} \in S_b} \sigma(\mathbf{r}_0) \nabla_0 G(\mathbf{r}; \mathbf{r}_0) \ , \end{aligned}\]

having defined vortex intensity \(\boldsymbol\gamma(\mathbf{r}) := - \hat{\mathbf{n}} \times \nabla \mu(\mathbf{r})\), and the source intensity \(\sigma(\mathbf{r}) := \hat{\mathbf{n}}(\mathbf{r}) \cdot \nabla \varphi(\mathbf{r})\).

Circulation along a line \(\ell_0\) is defined as

\[\Gamma_{\ell_0} = \oint_{\ell_0} \mathbf{u}'(\mathbf{r}_0) \cdot \hat{\mathbf{t}}(\mathbf{r}_0) \ ,\]

Focusing on 2-dimensional problems, the surface \(S_b\) becomes a line \(\ell_b\). Let’s deal here with closed surfaces: thus the integral over closed surface (line, in 2D) \(\ell_b\) becomes \(\oint_{\ell_b}\). The contribution to circulation of the source term is identically zero, as

\[\begin{split}\begin{aligned} \Gamma_{\ell_0}^{\sigma} & = \oint_{\mathbf{r}_0 \in \ell_0} \hat{\mathbf{t}}(\mathbf{r}_0) \cdot \oint_{\mathbf{r} \in \ell_b} \sigma(\mathbf{r}) \nabla_0 G(\mathbf{r}; \mathbf{r}_0) = \\ & = \oint_{\mathbf{r} \in \ell_b} \sigma(\mathbf{r}) \oint_{\mathbf{r}_0 \in \ell_0} \hat{\mathbf{t}}(\mathbf{r}_0) \cdot \nabla_0 G(\mathbf{r}; \mathbf{r}_0) = \\ & = \oint_{\mathbf{r} \in \ell_b} \sigma(\mathbf{r}) \oint_{\mathbf{r}_0 \in \ell_0} d_0 G(\mathbf{r}; \mathbf{r}_0) = \\ & = 0 \ . \end{aligned}\end{split}\]

Thus, the contribution to circulation comes from vortex term only. Manipulating this contribution, it’s possible to get an expression for the circulation as a function of the vortex intensity,

\[\begin{split}\begin{aligned} \Gamma_{\ell_0} = \Gamma_{\ell_0}^{\gamma} & = \oint_{\mathbf{r}_0 \in \ell_0} \hat{\mathbf{t}}(\mathbf{r}_0) \cdot \oint_{\mathbf{r} \in \ell_b} \boldsymbol\gamma(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) = \\ & = - \oint_{\mathbf{r} \in \ell_b} \boldsymbol\gamma(\mathbf{r}) \cdot \oint_{\mathbf{r}_0 \in \ell_0} \hat{\mathbf{t}}(\mathbf{r}_0) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) = \\ \end{aligned}\end{split}\]

The integral over \(\ell_0\) is equal to \(-\hat{\mathbf{z}}\) (see below), and thus

\[\Gamma_{\ell_0} = \oint_{\mathbf{r} \in \ell_b} \gamma_z(\mathbf{r}) \ .\]

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

Method 1 - Using Green’s lemma. In a 2-dimensional plane, using Cartesian coordinates, \(\mathbf{r} = x \hat{\mathbf{x}} + y \hat{\mathbf{y}}\), and Green’s lemma in the plane on a simple domain, the integral becomes

\[\begin{split}\begin{aligned} \oint_{\mathbf{r} \in \ell} \hat{\mathbf{t}} \times \nabla G(\mathbf{r}; \mathbf{r}_0) & = \hat{\mathbf{z}} \oint_{\mathbf{r} \in \ell} \left\{ t_x \partial_y G - t_y \partial_x G \right\} = && \text{(Green's lemma)} \\ & = - \hat{\mathbf{z}} \int_{\mathbf{r} \in S_\ell} \left\{ \partial_{yy} G + \partial_{xx} G \right\} = \\ & = - \hat{\mathbf{z}} \int_{\mathbf{r} \in S_\ell} \nabla^2 G(\mathbf{r}; \mathbf{r}_0) = \\ & = 0 \ , \end{aligned}\end{split}\]

if the singular point \(\mathbf{r}_0\) of \(G(\mathbf{r}; \mathbf{r}_0)\) is outside \(S_{\ell}\), as \(-\nabla^2 G(\mathbf{r}; \mathbf{r}_0) = \delta(\mathbf{r}-\mathbf{r}_0)\), and \(\int_{\mathbf{r} \in S_\ell} \delta(\mathbf{r}_0 - \mathbf{r}) f(\mathbf{r}) = E_{S_\ell}(\mathbf{r}_0) f(\mathbf{r}_0)\).

In a non-simply connected domain, it’s possible to introduce a cut in the domain. If the cut is geometrical and not physical - i.e. w/o any discontinuity of the functions across it -, the value of the integral is independent from the path, given a path winding around the hole in the domain the same number of times (todo Add a link to some materia in (Not so) Mathematics Basics),

\[\oint_{\mathbf{r} \in \ell_1} \hat{\mathbf{t}} \times \nabla G(\mathbf{r}; \mathbf{r}_0) = \oint_{\mathbf{r} \in \ell_2} \hat{\mathbf{t}} \times \nabla G(\mathbf{r}; \mathbf{r}_0) \ .\]

Exploiting this path-independence, the value of the integral can be evaluated on a circle encolsing \(\mathbf{r}_0\),

\[\begin{split}\begin{aligned} \oint_{\mathbf{r} \in \ell_1} \hat{\mathbf{t}} \times \nabla G(\mathbf{r}; \mathbf{r}_0) & = \oint_{\mathbf{r} \in C} \hat{\mathbf{t}} \times \nabla G(\mathbf{r}; \mathbf{r}_0) = \\ & = \int_{\theta = 0}^{2 \pi} \hat{\boldsymbol\theta} \times \frac{1}{2 \pi} \frac{\hat{\mathbf{r}}}{|\mathbf{r}|} |\mathbf{r}| d \theta = \\ & = - \hat{\mathbf{z}} \ . \end{aligned}\end{split}\]

Asymptotic behavior of the perturbation potential

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 perturbation velocity reads

\[\begin{split}\begin{aligned} \mathbf{u}'(\mathbf{r}_0) & = \int_{\mathbf{r}\in S_b} \boldsymbol\gamma(\mathbf{r}) \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0) + \int_{\mathbf{r} \in S_b} \sigma(\mathbf{r}) \nabla_0 G(\mathbf{r}; \mathbf{r}_0) = \\ & \simeq \int_{\mathbf{r}\in S_b} \boldsymbol\gamma(\mathbf{r}) \times \left[ \frac{1}{2 \pi} \frac{\mathbf{r}_0}{|\mathbf{r}_0|^2} - \frac{1}{2 \pi} \frac{\mathbf{r}}{|\mathbf{r}_0|^2} + \frac{1}{\pi} \mathbf{r} \cdot \mathbf{r}_0 \frac{\mathbf{r}_0}{|\mathbf{r}_0|^4} \right] + \\ & \quad + \int_{\mathbf{r} \in S_b} \sigma(\mathbf{r}) \left[ \frac{1}{2 \pi} \frac{\mathbf{r}_0}{|\mathbf{r}_0|^2} - \frac{1}{2 \pi} \frac{\mathbf{r}}{|\mathbf{r}_0|^2} + \frac{1}{\pi} \mathbf{r} \cdot \mathbf{r}_0 \frac{\mathbf{r}_0}{|\mathbf{r}_0|^4} \right] \ . \end{aligned}\end{split}\]

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 first term in the first integral is proportional to the «vector» circulation \(\boldsymbol\Gamma = \Gamma \hat{\mathbf{z}} = \oint_{\mathbf{r} \in \partial S_b} \boldsymbol\gamma(\mathbf{r})\) (todo Prove it), thus the asymptotic behavior of the perturbation velocity as \(|\mathbf{r}_0| \gg |\mathbf{r}|\) reads

\[\mathbf{u}'(\mathbf{r}_0) \simeq \frac{1}{2 \pi} \boldsymbol\Gamma \times \frac{\mathbf{r}_0}{|\mathbf{r}_0|^2} + \frac{1}{2 \pi} \dot{m} \frac{\mathbf{r}_0}{|\mathbf{r}_0|^2} + O\left( \frac{1}{|\mathbf{r}_0|^2} \right) \]

8.3.5.2.2. 3-dimensional problems

Green’s function Green’s function of a 3-dimensional Poisson problem reads

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

Solution

8.3.5.3. Wake and the shape of the domain

Physical conditions, with jump conditions

No physical wake in 2-dimensional steady flows

Wake in steady 2-dimensional flows

In 2-dimensional steady flows there’s no physical wake, i.e. there’s no jump in physical quantities like velocity or pressure across it. Anyways, for lifting bodies, for which \(\Gamma \ne 0\) (see Kutta-Joukowski theorem), there’s the need to introduce a cut/branch in the domain in order to geta single-valued potential function \(\varphi\).

…

Evaluating the circulation along a path winding around the airfoil, expressing the perturbation velocity as the gradient of the perturbation potential, a relation between the circulation and the potential jump across the wake is readily found

\[\Gamma = \oint_{\ell} \mathbf{u} \cdot \hat{\mathbf{t}} = \oint_{\ell} \hat{\mathbf{t}} \cdot \nabla \varphi = \int_{w^+}^{w^-} d \varphi = \varphi_{w^+} - \varphi_{w^-} = \Delta \varphi_w \ .\]

8.3.5.4. Theorems

8.3.5.4.1. Kutta-Joukowski theorem

2-dimensional flows. Circulation along a path \(\gamma\) is defined as the integral

\[\Gamma = \oint_{\gamma} \mathbf{u} \cdot \hat{\mathbf{t}} \ .\]

Let’s define here a circuilation vector \(\boldsymbol\Gamma = \Gamma\hat{\mathbf{z}}\), with \(\hat{\mathbf{z}}\) orthogonal w.r.t. the plane of the 2-dimensional flow.

Using integral balance equations of mass and momentum

\[\begin{split}\begin{aligned} & \frac{d}{dt} \int_V \rho + \oint_{\partial V} \rho \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = 0 \\ & \frac{d}{dt} \int_V \rho \mathbf{u} + \oint_{\partial V} \rho \mathbf{u} \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = \int_V \rho \mathbf{g} + \oint_{\partial V} \mathbf{t}_{\hat{\mathbf{n}}} \ , \end{aligned}\end{split}\]

under the assumptions of steady flow and control volume, \(\frac{d}{dt} \equiv 0\) and \(\mathbf{u}^{rel} = \mathbf{u}\), negligible volume force «\(\mathbf{g} \sim \mathbf{0}\)», and negligible viscous stress, \(\mathbf{t}_{\hat{\mathbf{n}}} = - p \hat{\mathbf{n}}\), and rearranging the surface integrals of stress to get the force \(\mathbf{F}\) acting on solid body \(S_b\), it follows

\[\begin{split}\begin{aligned} & \oint_{\partial V} \rho \mathbf{u} \cdot \hat{\mathbf{n}} = 0 \\ & \oint_{\partial V} \rho \mathbf{u} \mathbf{u} \cdot \hat{\mathbf{n}} = - \mathbf{F} + \oint_{S_{\infty}} \mathbf{t}_{\hat{\mathbf{n}}} \ . \end{aligned}\end{split}\]

Details

…

The velocity field as the sum of the free-stream velocity and the perturbation velocity,

\[\mathbf{u} = \mathbf{U}_{\infty} + \mathbf{u}' = \mathbf{U}_{\infty} + \nabla \varphi \ .\]

Under the additional assumption of uniform free-stream velocity, zero vorticity follows and the second Bernoulli theorem holds,

\[\begin{split}\begin{aligned} P_{\infty} + \frac{1}{2} \rho U_{\infty}^2 & = P + \frac{1}{2} \rho |\mathbf{u}|^2 = \\ & = P + \frac{1}{2} \rho U_\infty^2 + \rho \mathbf{U}_\infty \cdot \mathbf{u}' + \frac{1}{2} \rho |\mathbf{u}'|^2 \ , \end{aligned}\end{split}\]

being \(P\), \(\mathbf{u}\) the pressure and velocity in any point of the fluid that is reached by a streamline coming from the free-stream. The expression of the aerodynamic force acting on the body can be recast as

\[\begin{split}\begin{aligned} \mathbf{F} & = - \oint_{S_\infty} \rho \mathbf{u} \mathbf{u} \cdot \hat{\mathbf{n}} - \oint_{S_\infty} P \hat{\mathbf{n}} = \\ & = - \underbrace{\oint_{S_\infty} \rho \mathbf{U}_\infty \mathbf{U}_\infty \cdot \hat{\mathbf{n}}}_{ = \mathbf{0} \ , \ \ \oint_{S} \mathbf{const} \cdot \hat{\mathbf{n}} = 0 } - \underbrace{\oint_{S_\infty} \rho \mathbf{U}_\infty \mathbf{u}' \cdot \hat{\mathbf{n}}}_{ = \mathbf{0} \text{ , from mass balance}} - \oint_{S_\infty} \rho \mathbf{u}' \mathbf{U}_\infty \cdot \hat{\mathbf{n}} - \underbrace{\oint_{S_\infty} \rho \mathbf{u}' \mathbf{u}' \cdot \hat{\mathbf{n}}}_{ \rightarrow \ \mathbf{0} \text{ as } |S_{\infty}| \rightarrow \ \infty } - \oint_{S_\infty} P \hat{\mathbf{n}} = \\ & \simeq - \oint_{S_\infty} \rho \mathbf{u}' \, \mathbf{U}_\infty \cdot \hat{\mathbf{n}} - \underbrace{\oint_{S_\infty} P_\infty \hat{\mathbf{n}}}_{= \mathbf{0} \ , \ \ \oint_{S} \hat{\mathbf{n}} = \mathbf{0}} + \oint_{S_\infty} \rho \mathbf{U}_\infty \cdot \mathbf{u}' \ \hat{\mathbf{n}} + \underbrace{\oint_{S_\infty} \frac{1}{2} \rho |\mathbf{u}'|^2 \, \hat{\mathbf{n}}}_{ \rightarrow \ \mathbf{0} \text{ as } |S_{\infty}| \rightarrow \ \infty } = \\ & \simeq - \rho \mathbf{U}_\infty \cdot \oint_{S_\infty} \left( \hat{\mathbf{n}} \, \mathbf{u}' - \mathbf{u}' \, \hat{\mathbf{n}} \right) = \\ & = \rho \mathbf{U}_\infty \times \boldsymbol\Gamma \ . \end{aligned}\end{split}\]

Circulation in Kutta-Joukowski theorem

\[\begin{split}\begin{aligned} \mathbf{U}_\infty \times \boldsymbol\Gamma & = \mathbf{U}_\infty \times \hat{\mathbf{z}} \oint_{S} \mathbf{u} \cdot \hat{\mathbf{t}} = \\ & = \mathbf{U}_\infty \times \hat{\mathbf{z}} \oint_{S} \left( \mathbf{U}_\infty + \mathbf{u}' \right) \cdot \hat{\mathbf{t}} = \\ & = \mathbf{U}_\infty \times \hat{\mathbf{z}} \oint_{S} \mathbf{u}' \cdot \hat{\mathbf{t}} = \\ & = \mathbf{U}_\infty \times \hat{\mathbf{z}} \oint_{S} \mathbf{u}' \cdot \left( \hat{\mathbf{z}} \times \hat{\mathbf{n}} \right) \ , \end{aligned}\end{split}\]

Using a Cartesian basis, the \(i^{th}\) component of the vector reads

\[\begin{split}\begin{aligned} \left\{ \mathbf{U}_\infty \times \boldsymbol\Gamma \right\}_i & = \varepsilon_{ijk} U_j \delta_{kz} \oint_{S} u'_l \varepsilon_{lpq} \delta_{zp} n_q = \\ & = \varepsilon_{ijz} \varepsilon_{qlz} U_j \oint_{S} u'_l n_q = \\ & = \left( \delta_{iq} \delta_{jl} - \delta_{il} \delta_{jq} \right) U_j \oint_{S} u'_l n_q = \\ & = U_j \oint_{S} \left( u'_j n_i - n_j u'_i \right) = \\ & = \left\{ - \mathbf{U}_\infty \cdot \oint_{S} \left( \hat{\mathbf{n}} \, \mathbf{u}' - \mathbf{u}' \hat{\mathbf{n}} \right) \right\}_i \ . \end{aligned}\end{split}\]

having used the properties of the Ricci symbols, \(\varepsilon_{ijk} = \varepsilon_{jki}\), and the identity \(\varepsilon_{ijk} \varepsilon_{ilm} = \delta_{jl} \delta_{km} - \delta_{jm} \delta_{kl}\).

8.3.5.4.2. D’Alembert paradox - 2 dimensional flow

Lift and drag are defined as the components of the aerodynamic force orthogonal and parallel to the free-stream velocity \(\mathbf{U}_\infty = U_\infty \hat{\mathbf{x}}\). Being \(\mathbf{F}\) the aerodynamic foce acting on the body,

\[\begin{split}\begin{aligned} \mathbf{F} & = \rho \mathbf{U}_\infty \times \boldsymbol\Gamma = \\ & = \rho \left( U_\infty \hat{\mathbf{x}} \right) \times \left( \Gamma \hat{\mathbf{z}} \right) = \\ & = - \rho U_\infty \Gamma \hat{\mathbf{y}} \ , \end{aligned}\end{split}\]

the lift and drag are

\[\begin{split}\begin{aligned} L & = F_y = - \rho U_{\infty} \Gamma && \text{(Kutta-Joukowski theorem)} \\ D & = F_x = 0 && \text{(D'Alembert paradox)} \ . \end{aligned}\end{split}\]