## Numerical Background

The numerical method used in panMARE is a 3-dimensional zeroth-order panel method. The surface of a body is divided into quadrilateral body and wake panels. On every body panel a source with strength $$\sigma$$ and a dipole with strength $$\mu$$ is placed. On the wake only dipoles with strength $$\mu_{W}$$ are located. All strength are constant over one panel but can vary among each other.

In addition, on each panel a collocation point is defined. In panMARE the collocation point is defined as the midpoint of the panel. In the end, the discrete formulation of the boundary condition equations is the following:

$$\sum\limits_{i=1}^{N} \mu_{i} C_{i} + \sum\limits_{i=1}^{N_{W}}\mu_{w,i} C_{w,i} - \sum\limits_{i=1}^{N} \sigma_{i} B_{i} = 0$$ where $$C_{i}$$ and $$B_{i}$$ are influence functions which are defined as: $$\begin{split} C_{i}(x) & := \frac{1}{4 \pi} \int\limits_{S_{i}} (\frac{\partial}{\partial n} (\frac{1}{r})) dS, \quad \forall i=1, \dots, N \quad \text{and} \quad \forall i=1, \dots, N_{w}\\ B_{i}(x) & := \frac{1}{4 \pi}\int\limits_{S_{i}} (\frac{1}{r})dS \quad \forall i=1, \dots , N. \end{split}$$ We use the linear Kutta condition and define new dipole coefficients: $$\begin{split} A_{i}(x) = & \begin{cases} & C_{i}(x), \quad \text{if the panel does not lie on the trailing edge}\\ & C_{i}(x) \pm \sum\limits_{l=1}^{N_{i}}C_{l}(x), \quad \text{if the panel $$i$$ lies on the trailing edge}, \\ & \qquad \qquad \qquad \qquad \qquad \text{ $$+$$ suction side, $$-$$ pressure side.} \end{cases} \end{split}$$ Applying this equation on each collocation point of the body leads to a system of $$N\times N$$ equations for the $$N$$ unknown dipole strengths $$\mu$$: $$\begin{pmatrix} A_{1,1} & \dots & A_{1,N} \\ \vdots & \vdots & \vdots \\ A_{N,1} & \cdots & A_{N,N} \end{pmatrix} \cdot \begin{pmatrix} \mu_{1} \\ \vdots \\ \mu_{N} \end{pmatrix} = \begin{pmatrix} B_{1,1} & \dots & B_{1,N} \\ \vdots & \vdots & \vdots \\ B_{N,1} & \cdots & B_{N,N} \end{pmatrix} \cdot \begin{pmatrix} \sigma_{1} \\ \vdots \\ \sigma_{N} \end{pmatrix}.$$

## Wake Modeling

In the steady calculations, the strength of the trailing panels does not vary with time but are constant for all panels which originate from one panel at the trailing edge. The location of the trailing wake panels is unknown at the beginning of the solution and has to be found in an iterative manner until it follows streamlines starting from the trailing edge.

For propeller steady flow calculations the algorithm is as follows:

• Solve the boundary value problem (BVP) with prescribed helical wake.
• Calculate the unknown dipole strength ($$\mu$$) on the body surface and on the first wake strip.
• Calculate induced velocities ($$v_{ind}$$) by approximating the derivatives of the dipole strength.
• Find the new wake coordinates by aligning with the total local velocities: $$v_{total}=v_{ind}+V_{\infty}.$$
• Solve the BVP again with the updated wake geometry and align the wake geometry until the difference of the wake geometries between two iteration steps is within a given tolerance.
• Save the calculated wake geometry and strength values on the wake and on the body for the unsteady wake alignment process. These steady results are the initial values for the unsteady calculations.

For propeller unsteady flow calculation, first a steady flow calculation is performed. The time-stepping algorithm is as follows:

• Set initial wake shape which is the same as calculated in the steady mode.
• Solve the integral equations at each time step and calculate the unknown dipole strength ($$\mu)$$ on the body surface and on the first wake strip.
• Determine induced velocities ($$v_{ind}$$) from the calculated dipole strength.
• Compute the new position of the wake panels, the method is the same as in steady calculations.
• Solve the BVP again with the updated wake geometry.
• Move to the next time step and align the wake geometry.
• Return to step 2 and repeat the procedure for the new time step until the difference of the wake geometries between two iteration steps is within a given tolerance.

Figure 1: Wake Alignment.

## Remarks

Panel methods are used for flow calculations where the viscous effects can be neglected or calculated by other methods. The usage of potential based boundary element methods like the panel method is widely spread in the propeller design due to their short calculation time. A three-dimensional problem is namely reduced to a two-dimensional problem (Boundary Element Problem).