# Numerical Background

This page provides an overview of the numerical implementations used to model hydrofoils and wake in order to calculate the flow around a propeller. You can find additional information on the numerical background of *pan*MARE on the pages of individual *pan*MARE techniques and in various *pan*MARE-related publications of the Institute for Fluid Dynamics and Ship Theory.

##### Numerical Implementation

The flow field is calculated by a low-order panel method in *pan*MARE.

The body surface \(S_B\) is discretised with panels with constant doublet and source strengths, while only doublet strengths are used for the wake sheet \(S_W\). The panels are mainly quadrilateral and planar. An example of a hydrofoil's discretisation is provided in Figure 2. Hereby, the hydrofoil is modelled using \(N_B\) panels. The wake sheet is connected to the hydrofoil's trailing edge with \(N_W\) wake panels and oriented according to the flow. Each body panel moves through the fluid domain with a velocity \(\vec{v}_{\text{mot}}\).

It is assumed that the doublet strength \(\mu\) and the source strength \(\sigma\) are constant along a respective panel. Thus the following summation formulation of the velocity potential is possible: $$\Phi = \sum_j^{N_B}\left[\left(\frac{1}{4\pi} \int_{S_j} \left[\frac{\partial}{\partial n}\left(\frac{1}{r}\right)\right]~\text{d}S\right)\mu_j - \left(\frac{1}{4\pi} \int_{S_j} \left[\left(\frac{1}{r}\right)\right]~\text{d}S\right)\sigma_j \right]$$ $$~~~~~+ \sum_j^{N_W}\left[\left(\frac{1}{4\pi} \int_{S_j} \left[\frac{\partial}{\partial n}\left(\frac{1}{r}\right)\right]~\text{d}S\right)\mu_j \right] + \Phi_{\text{ext}}$$ Hereby, the first sum is of all body panels \(N_B\) and the second sum is of all wake panels \(N_W\).

A collocation point is assigned to each panel, which is located at the panel's centre. For the body panels the collocation point is positioned on the inside of the body. The collocation point is the point at which the boundary conditions are evaluated. Thus an equation system is set in order to calculate the influence of the doublet \(C_{i,j}\) resp. \(C_{i,k}\) or the source \(B_{i,j}\) of each body panel \(j\) respectively wake panel \(k\) on the collocation point of each body panel \(i\). $$ \begin{split} C_{i,j} &= \frac{1}{4\pi} \int_{S_j} \left[\frac{\partial}{\partial n}\left(\frac{1}{r}\right)\right]~\text{d}S\\ C_{i,k} &= \frac{1}{4\pi} \int_{S_k} \left[\frac{\partial}{\partial n}\left(\frac{1}{r}\right)\right]~\text{d}S\\ B_{i,j} &= - \frac{1}{4\pi} \int_{S_j} \left[\left(\frac{1}{r}\right)\right]~\text{d}S\\ \end{split} $$

The integrals in the equations above can be calculated analytically. According to the Morino-Kutta condition, the doublet strength on the wake panels equals the doublet strength difference of the body panels at the trailing edge. Therefore, it is possible to unite the coefficients \(C_{i,j}\) and \(C_{i,k}\) and substitute the body panel's coefficients. Hereby, steady cases are distinguished from unsteady cases.

##### Steady Case

In case of steady flow, the doublet strength of the wake does not change in time. Therefore, all wake panels which are dispatched from a body panel at the trailing edge have the same doublet strength. Furthermore, the wake maintains its orientation in regard to the body (except for its shape). $$ A_{\text{steady},i,j} = \left\{\begin{matrix} C_{i,j} \pm \sum_K C_{i,k} & \left\{\begin{matrix} \forall j \in \text{body panels at the trailing edge with dispatching wake}, \\ k \in \text{of $j$ dispatching wake panels}, \\ ~~~~~~\text{with \(+\) suction side, \(-\) pressure side.}\\ \end{matrix}\right. \\ C_{i,j} & \forall j \in \text{body panels without dispatching wake} \end{matrix}\right. $$ This way the summation formulation of the velocity potential can be evaluated at every collocation point on the body as follows: $$ \Phi_i = \sum_j^{N_B}\left[C_{i,j}\mu_j - B_{i,j}\sigma_j \right] + \sum_k^{N_W}\left[C_{i,k}\mu_k \right] + \Phi_{\text{ext},i} $$ By applying the Neumann boundary condition introduced above, the doublet strength \(\sigma_i\) of every body panel can be calculated at the collocation point. Using the Dirichlet boundary condition (\(\Phi_i-\Phi_{\text{ext},i}=0\)) subsequently, the following system of equations is established: $$ \sum_j^{N_B}\left[A_{\text{steady},i,j}\mu_j\right] = - \sum_j^{N_B}\left[B_{i,j}\sigma_j \right] $$ Based on this, the body panels' doublet strengths can be calculated directly, and the wake panels' doublet strengths can be derived from the differences between the trailing edge body panels' doublet strengths.

##### Unsteady Case

In unsteady cases, the body moves according to the specified body motion. While the body panels are moving, the wake panels remain at fixed positions. Thus, the body is moving away from the wake panels in the unsteady calculation. For this reason, the wake is detatched from the body during each time step. The wake is detatched just before the body moves. After the wake is detatched, new wake panels are inserted in the resulting gap between the hydrofoil's trailing edge and the old panels. The "length" of these new panels is set corresponding to the time increment. In order to consider the time rate of change of the wake's doublet strength, only the first wake panels at the trailing edge are calculated and all remaining wake panels maintain their values from previous time steps. This results in the following readily changing coefficients: $$ A_{\text{unsteady},i,j} = \left\{\begin{matrix} C_{i,j} \pm C_{i,k} & \left\{\begin{matrix} \forall j \in \text{body panels at the trailing edge with dispatching wake}, \\ k = \text{dispatching wake panels attached to $j$}, \\ ~~~~~~\text{with \(+\) suction side, \(-\) pressure side.}\\ \end{matrix}\right. \\ C_{i,j} & \forall j \in \text{body panels without dispatching wake} \end{matrix}\right. $$ Same as in the steady case, the coefficients above are used for the evaluation of the velocity potential's summation formulation. The Neumann boundary condition is used to calculate the douplet strength \(\sigma_i\) of each body panel at its collocation point. Thereafter, the Dirichlet boundary condition is used to establish the following system of equations: $$ \sum_j^{N_B}\left[A_{\text{unsteady},i,j}\mu_j\right] = - \sum_j^{N_B}\left[B_{i,j}\sigma_j \right] - \sum_k^{N_W}\left[\left\{\begin{matrix} C_{i,k}\mu_k & \forall k \in \text{old wake panels} \\ 0 & \forall k \notin \text{old wake panels} \end{matrix}\right.\right] $$ Hereby, the influence of the new wake panels is substituted in \(A_{\text{unsteady},i,j}\), and the old panels' influences and doublet strengths are part of the right hand side of the equation, since they are known values. Same as in the steady case, the body panels' doublet strengths can be calculated directly, based on this, whereas the wake panels' doublet strengths can again be derived from the differences between the trailing edge body panels' doublet strengths.

##### Induced Potential and Pressure Calculation

In both steady and unsteady cases the determined doublet strengths are used to calculated the induced potential \(\Phi_{\text{ind}}\) at the body panels' collocation points. The induced potential is equal to the negative doublet strength \(\mu\): $$ \Phi_{\text{ind},i} = - \mu_i $$ The local flow velocity \(\vec{v}_\text{local}\) at the panels is determined relative to the panels. It is calculated using the gradient of the local perturbations of the underlying currents' potential (i.e. undisturbed potential) \(\Phi_{\text{ext}}\) and the induced potential \(\Phi_{\text{ind}}\). Additionally, the time rate of change of the potential \(\Phi\), also relative to the panels, is calculated via finite differences. Based on the determined values, it is possible to evaluate Bernoulli's equation and thus determine the pressure acting on the body panels: $$ p = \rho \left[ \vec{g} \cdot \left( \vec{P}_{\infty} - \vec{P} \right) - \frac{1}{2} \left(\nabla \Phi_{\text{ind}} + \nabla \Phi_{\text{ext}} - \vec{v}_{\text{mot}}\right)^2 - \frac{\partial \left( \Phi_{\text{ind}} + \Phi_{\text{ext}} \right)}{\partial t} + \frac{1}{2} \vec{v}_{\text{mot}}^2 \right] + p_{\infty} $$ The pressure on the body panels includes the reference pressure \(p_{\infty}\) at a reference point \(\vec{P}_{\infty}\) in the fluid.

##### Wake Sheet Orientation

In addition to the solution of the flow problem and the calculation of the flow velocities and acting pressure, the orientation of the hydrofoil's wake sheet is realigned to follow the lines of the flow. The wake sheet's initial orientation is set corresponding to the hydrofoil's motion at the start of the calculation. Hereby, a helical shape is initialised in cases of rotating bodies (e.g. propeller).

Theoretically, the wake sheet's length should be infinite. However, since this is not feasible from a numerical point of view, the wake sheet is modelled with a sufficiently large finite length. The modelled wake sheet's length is chosen in a way that the hydrofoil is not influenced significantly by the vortex at the sheet's end. This criterion is also applied whenever the wake sheet is shortened in unsteady cases, after new wake panels have been inserted.

During the calculation, the wake panels are oriented corresponding to the induced velocities and the gradient of the local perturbation of the underlying currents' potential. This way, the wake panels are oriented along the flow lines.

It should be noted that a large number of wake panels can lead to very long computing times. However, the computational effort can be reduced significantly, if a predefined shape of the wake sheet is used.