# Theoretical Background

This page gives a short introduction of *pan*MARE's theoretical background. Thereby, *pan*MARE is based on the principles and equations of the potential method described in detail by Katz and Plotkin in *Low-Speed Aerodynamics*. More information can also be found in the Institute for Fluid Dynamics and Ship Theory's various *pan*MARE-related publications.

##### Potential Theory

The panel method *pan*MARE uses potential theory to calculate the flow.
A shematic representation of the problem to be solved is visualised in Figure 1, including
the boundaries and used nomenclature. One or more bodies with the surface \(S_B\)
are located in the fluid \(V\), which is enclosed by a domain boundary \(S_\infty\). It is
possible to further include boundaries such as walls, a free surface \(S_{B_{\text{FS}}}\),
or the seabed \(S_{B_{\text{G}}}\). These boundaries are counted among the fluid domain's
surrounding outer boundaries. The surface normal \(\vec{n}\) points into the fluid domain \(V\).
The body (or bodies) move(s) through the fluid domain, whereby translations, rotations and
deformations are possible in arbitrary combinations. The velocities resulting from the body-movement
are known at every point on the surface as \(v_{\text{mot}}\). For lift-generating bodies an additional
wake surface \(S_W\) is introduced, which prevents the flow around the hydrofoil's trailing edge, thus
satisfying the Kutta boundary condition.

Since the flow is assumed to be inviscid, irrotational and incompressible in potential theory, the conservation equations can be simplified resulting in the incompressible and irrotational equation of continuity (LaPlace equation) formulated for the velocity potential \(\Phi\): $$\Delta\Phi = \nabla^2\Phi = 0$$

and the Bernoulli equation for the pressure \(p\): $$\frac{p}{\rho} + \vec{g} \cdot \vec{P} + \frac{1}{2} \vec{v}^2 + \frac{\partial \Phi}{\partial t} = \text{const}.$$ Hereby, \(\Phi\) is the potential, whose spatial gradient \(\nabla\Phi\) corresponds to the flow velocity, whereas \(\vec{g}\) is the gravitational acceleration, \(\vec{P}\) the observation point's position and \(\vec{v}\) the flow velocity. Assuming that there is no flow through the body (boundaries \(S_B\) and \(S_W\)) and that the influence of the body vanishes at infinity (boundary \(S_\inf\)), a Green's identity can be used as a solution of the LaPlace equation: $$\int_{S}\left(\Phi_1\nabla\Phi_2-\Phi2\nabla\Phi_1\right)\cdot\vec{n}~\text{d}S = \int_{V}\left(\Phi_1\nabla^2\Phi_2-\Phi_2\nabla^2\Phi_1\right)~\text{d}V$$

The fluid domain's boundary \(S\) consists of all boundaries \(S_B\), \(S_W\) and \(S_\inf\) of the fluid \(V\). The identity equation above can be rearranged and the following subsitution of a doublet strength and a source strength applied: $$- \mu = \Phi - \Phi_I $$ $$ \sigma = \frac{\partial \Phi}{\partial n} - \frac{\partial \Phi_I}{\partial n}$$

Hereby, the doublet strength \(\mu\) corresponds to the jump of the velocity potential from \(\Phi_I\) within the body to \(\Phi\) outside the body in normal direction \(\vec{n}\), and the source strength \(\sigma\) corresponds to the jump of the velocity potential's gradient, both at the boundary \(S\). The normal direction is positive pointing away from the body, i.e. into the fluid. Based on this, the velocity potential at a point \(\vec{P}\) in the fluid can be described by the following equation: $$\Phi = \frac{1}{4\pi} \int_{S_B} \left[\mu\frac{\partial}{\partial n}\left(\frac{1}{r}\right) - \sigma\left(\frac{1}{r}\right)\right] ~\text{d}S + \frac{1}{4\pi} \int_{S_W} \left[\mu\frac{\partial}{\partial n}\left(\frac{1}{r}\right)\right] ~\text{d}S + \Phi_{\infty}$$

Hereby, \(r\) denotes the distance between the point of observation and the surface. For the wake surface there is no jump in the velocity potential's gradient in normal direction, since the wake surface is not a body. The fluid's perturbation caused by the body is represented by the potential. The undisturbed potential \(\Phi_\inf\) describes the condition of the fluid without the body's presence and includes possible currents as well as the influence of the distant boundary \(S_\inf\). Since the distant boundary has no contribution and its influence vanishes at infinity, the undisturbed potential \(\Phi_\inf\) is equal to the velocity potential \(\Phi_{\text{ext}}\) of the underlying currents, which is defined continuously throughout the fluid domain. Assuming that: $$\Phi_2 = \frac{1}{r}~,~~~\Phi_I = 0$$ and substituting \(\Phi_1\) with \(\Phi\) in the Green's identity, the velocity potential at a point \(\vec{P}\) in the fluid can be written as follows: $$\Phi = \frac{1}{4\pi} \int_{S_B} \left[\mu\frac{\partial}{\partial n}\left(\frac{1}{r}\right) - \sigma\left(\frac{1}{r}\right)\right] ~\text{d}S + \frac{1}{4\pi} \int_{S_W} \left[\mu\frac{\partial}{\partial n}\left(\frac{1}{r}\right)\right] ~\text{d}S + \Phi_{\text{ext}}$$ Thereby, the influence of the body is represented by the so-called induced potential \(\Phi_\text{ind}\). The spatial gradient of the induced potential corresponds to the induced velocity of the flow. $$\text{with:}~\Phi_{\text{ind}} = \frac{1}{4\pi} \int_{S_B} \left[\mu\frac{\partial}{\partial n}\left(\frac{1}{r}\right) - \sigma\left(\frac{1}{r}\right)\right] ~\text{d}S + \frac{1}{4\pi} \int_{S_W} \left[\mu\frac{\partial}{\partial n}\left(\frac{1}{r}\right)\right] ~\text{d}S$$ $$\Rightarrow \Phi = \Phi_{\text{ind}} + \Phi_{\text{ext}}$$

##### Boundary Conditions

To determine the doublet strength \(\mu\) and the source strength \(\sigma\) boundary conditions are required. Since the induced potential vanishes inside the body, the inner potential \(\Phi_I\) is defined as: $$\Phi_I = \left[\Phi_{\text{ind}} + \Phi_{\text{ext}}\right]_I = \Phi_{\text{ind},I} + \Phi_{\text{ext}} = \Phi_{\text{ext}}$$ This leads to the following formulation of the Dirichlet problem: $$\frac{1}{4\pi} \int_{S_B+S_W} \left[\mu \frac{\partial}{\partial n}\left(\frac{1}{r}\right)\right]~\text{d}S - \frac{1}{4\pi} \int_{S_B} \left[\sigma \left(\frac{1}{r}\right)\right]~\text{d}S = 0$$ This equation is evaluated at the inner side of the body's boundary \(S_B\). If the equation is evaluated at the distant boundary \(S_\inf\), the terms in the integrals vanish, so that the boundary condition is also fulfilled there.

Another boundary condition is that there must be no flow through the body (Neumann boundary condition). To satisfy the Neumann boundary condition the velocities in normal direction must vanish. The sum of the induced and the external potential's gradients and the body's motion velocity \(\vec{v}_{\text{mot}}~\) is calculated and multiplied with the normal vector. The result is set to zero. $$0 = \left[\nabla\left(\Phi_{\text{ind}} + \Phi_{\text{ext}}\right) - \vec{v}_{\text{mot}}\right] \cdot \vec{n}$$ This leads to the following relation: $$\frac{\partial\Phi_{\text{ind}}}{\partial n} = \vec{v}_{\text{mot}} \cdot \vec{n} - \frac{\partial\Phi_{\text{ext}}}{\partial n}$$ The contribution of the motion velocity \(\vec{v}_{\text{mot}}~\)is included as negative, since the body is moved throught the resting fluid. As explained above, the source strength \(\sigma\) is defined as the jump of the potential's gradient. Since both the gradient as well as the induced potential vanish inside the body, the source strength is: \(\sigma = \frac{\partial \Phi_{\text{ind}}}{\partial n}\). Thus the source strength can be written as follows: $$\sigma = \frac{\partial \Phi_{\text{ind}}}{\partial n} = \left(\vec{v}_{\text{mot}} - \vec{v}_{\text{ext}}\right) \cdot \vec{n}$$

##### Hydrofoil Wake Surface

For the calculation of a hydrofoil a wake surface \(S_W\) is is inserted at the trailing edge of the hydrofoil, see Figure 1, in order to prevent the fluid from flowing around the trailing edge. This wake surface can be viewed as the free vortex sheet. The Kutta boundary condition demands that the pressure difference between suction and pressure side vanishes at the trailing edge. Linearisation leads to the following Morino-Kutta condition: $$\mu_W = \mu_{B,\text{upper}} - \mu_{B,\text{lower}}$$ This equation establishes a connection between the doublet strengths on the upper and lower side of the trailing edge and the wake surface. As long as the flow is mainly perpendicular to the trailing edge, the linearisation does not lead to significant mistakes. The orientation of the free vortex sheet is parallel to the respective flow lines, since there is no flow through this surface.