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Theoretical Background

The simulation tool panMARE is a three-dimensional first-order panel method which is based on the traditional potential theory for irrotational fluids. The basic theoretical concept of potential theory used in panMARE is the following:

  • The fluid is assumed to be irrotational, inviscid and incompressible. With this assumptions the continuity equation results in the Laplace's equation for the fluid potential: $$\triangle \Phi^{*} = \nabla^{2} \Phi^{*} = 0, \forall \, (x,y,z)\in V$$ where \(V\) is the fluid domain and \(\Phi^{*} = \Phi + \Phi_{\infty}\) is the fluid potential. \(\Phi\) is the disturbed potential and \(\Phi_{\infty}\) is the free stream potential. The momentum equation results in the Bernoulli equation for the pressure:

    $$ p + \rho g z + \frac{1}{2} \rho \vert V \vert ^{2} + \rho \frac{\partial \Phi}{\partial t} = const$$ where \(\rho \) is the fluid density, \( V \) is the total fluid velocity and \( g \) is a gravitation constant.

  • The solution of the Laplace's equation is a linear combination of several sources and dipoles. In order to calculate the unknown source and dipole strengths a boundary element method is used. In an outer point \(\vec{x} \in \partial V\) of the boundary the solution is defined by: $$ \Phi(x,y,z) = \frac{1}{4 \pi} \int\limits_{\partial V} \mu \frac{\partial}{\partial n} (\frac{1}{r}) dS - \frac{1}{4 \pi} \int\limits_{\partial V} \sigma (\frac{1}{r}) dS $$ where \(\mu:=\Phi\), \( \sigma := \frac{ \partial \Phi}{\partial n}\) are the dipole and the source strength, respectively.

  • On the surface the potential is described by the Neumann boundary condition which states that the velocity components normal to the body's surface must vanish: $$\nabla \Phi^{*} \cdot \vec{n} = 0, \, \text{on the boundary} \, \partial V \quad \rightarrow \quad \sigma = \frac{ \partial \Phi}{\partial n}$$

  • On the wake of a lifting body the dipole strength is described by the Kutta condition which claims that at the trailing edge of a lifting body the pressure difference must vanish: $$ \triangle p_{TE} (\mu_{TE}) =0.$$ The above equation is nonlinear in \( \mu_{TE} \). A linear form of the Kutta condition is: $$ \mu_{TE} = \mu_{upper} - \mu_{lower}. $$

With the above assumptions a set of boundary condition equations is set up and solved in order to find the source and dipole strengths on the body surface and where necessary the dipole strength on the wake. From the calculated strengths the locally induced velocity components along the body's surface can be computed and the pressure can be determined by the Bernoulli equation. The induced velocities are calculated by:

$$v_{\xi}(x): =-\frac{\partial \mu}{\partial \xi} \qquad v_{\eta}(x): =-\frac{\partial \mu}{\partial \eta}$$ where \(\xi\) and \(\eta\) are the local tangential coordinates.