A low order panel method is used to predict the performance of propellers. A wake alignment model based on a pseudounsteady scheme is proposed and implemented. The results from this full wake alignment (FWA) model are correlated with available experimental data, and results from RANS for some propellers at design and low advance ratios. Significant improvements have been found in the predicted integrated forces and pressure distributions.

For the prediction of propeller performance, the panel method shows great advantage over other numerical tools. On one hand, the panel method recovers the real blade geometry as much as possible, instead of simplifying the propeller blades into lifting surfaces as the Vortex lattice method (VLM) does. On the other hand, the panel method is far more efficient than Reynolds-averaged Navier-Stokes (RANS) simulation, not only in a computational sense, but also on grid generation. Generally, the panel method strikes a good balance on accuracy and computational efficiency. Close to the design condition, the results from panel method are in good agreement with either experimental measurement or RANS simulation.

As the advance ratio becomes smaller, the pitch of the wake sheets becomes smaller. Consequently, the wake shed from one blade could be close to the consequent blade and, thus, strongly affect the pressure distribution on the consequent blade. Therefore, in order to correctly predict the propeller performance at low advance ratios, an accurate wake model is necessary. Intensive investigation has been taken in the past two decades on modeling the wake of a propeller.

Greeley and Kerwin [

Independently, Politis [

In this paper, the same idea as that in He’s [

The velocity flow field is decomposed into two components: inflow velocity and perturbation velocity, written as follows:

The methodology applied in this study is based on the inviscid potential flow theory:

Applying the kinematic boundary condition to the solid surfaces, we obtain the following:

At trailing edge,

For steady state, (

For unsteady problem in 2D, or even steady problem in 3D, (

The local perturbation velocities in the computational domain are not required for evaluating the pressure on the blade surface, but evoked by the wake model. Theoretically, the local perturbation velocity

If the field point

However, the velocities are finite at the control points only if the panels are regularly and smoothly distributed in space. At the late stage of the unsteady free wake evolution, or in certain intermediate steps to determine the steady wake, the geometry of the vortex sheet could be entangled. The velocity evaluated at the control point of a panel could lead to a huge deviation from the exact value. Numerical tests show that evaluating the velocity at nodal point with certain desingularized vortex kernel can be more robust than the schemes mentioned above.

In 2D, the dipole-induced velocity in (

Krasny [

In 3D, the vortex sheet is discretized into constant dipole elements, which are equivalent to vortex loops with constant vorticity strength. Therefore, induced velocity due to the vortex loop can be evaluated using the Biot-Savart law:

Replacing the Biot-Savart kernel

The free vortex sheets shed from the trailing edge from a propeller blade are material surfaces, which have to be aligned with the local flow velocity. Alignment schemes play very important roles in determining the spatial geometry or distribution of the free vortex sheets. For the steady and unsteady cases, the alignment schemes are numerically different.

Engineering community is more interested in the steady propeller performances. As an analogy to the Euler-explicit scheme for the unsteady cases, many researchers align the vortex sheet using the velocity at the upstream nodal points:

Numerical tests show that this scheme is stable and converges fast. However, this scheme has the same deficiency as the Euler-explicit scheme for the unsteady alignment: it continuously enlarges the rotating radius around the center of rollingup, and thus cannot predict the correct location and size of the rolling-up region. Therefore, in order to get better prediction of the wake geometry, alignment scheme in a manner of trapezoidal rule is preferred. However, direct application of the trapezoidal rule turned out to be unstable.

Tian and Kinnas [

As shown in Figure

Schematic plot for the pseudounsteady approach of alignment.

We have

Discretizing (

Notice that (

Consider a point under cylindrical coordinates

Now the averaged perturbation velocity

Defining

When applying (

Before applying the FWA model, the basic framework of current panel code is first tested with a 3D wing at low angle of attack (AOA). A tip-swept wing, which is expected to resemble a propeller blade with a large skew angle, is investigated. As shown in Figure

Geometry of the wing and comparison of pressure distributions with RANS.

The results at low AOA (AOA=2DEG) from current BEM model is validated with RANS, which is obtained through commercial software STAR-CCM+.

Three sections are taken to compare the pressure distribution. As shown in Figures

The panel code is applied to a five-bladed propeller DTNSR 4381, which is investigated by Boswell [

For the simulations with the panel code, two wake models are adopted: PSF-2 wake model, which is a fast wake alignment scheme developed by Greeley and Kerwin, [

RANS simulations of the same propeller are carried out with commercial code FLUENT, for

In the experimental study by Boswell [

KT and 10KQ for propeller DTNSR 4381 from PROPCAV. Experimental data are extracted from [

KT and 10KQ for propeller DTNSR 4381 from PROPCAV with different number of spanwise panels by chordwise panels on the blade and different streamwise grid size on the wake, where 6DEG stands for 60 streamwise wake panels per turn, and 10DEG stands for 36 streamwise wake panels per turn.

Figure

Wake geometry from FWA at different advance ratio ((a)

Wake geometry from PSF-2 at different advance ratio ((a)

Sections on the aligned wake cut at

More detailed comparisons of the results between BEM and RANS are carried out in two different blade sections at two different advance ratios, as shown in Figure

Comparison of pressure from different numerical methods at

In the case of

Figure

Convergence history of the FWA model for Propeller 4381 at

It should be pointed out that when the advance ratio is low enough, the leading edge vortex (LEV) could be developed, and affect the pressure distribution and total integrated forces. Greeley [

The panel code in this study neither included the LEV model nor coupled with the boundary layer solver. The viscous effects are considered by using an empirical viscous pitch correction [

Jessup [

Figure

Radial location of tip vortex for propeller DTMB 4119 at design

KT and 10KQ predicted by the BEM/FWA model with different values of

More rational approachs to determine

In this paper, a low-order panel method is used to predict the performance of propellers. A wake alignment model based on a pseudounsteady scheme is presented. In order to validate the basic frame work of the panel code without the fully aligned wake model, the method is applied first to a 3D wing at low angle of attack. The predicted pressure distribution agrees well with the RANS results. In case of the propeller working at design condition, both PSF-2 and FWA wake models predict reasonable KT and KQ. The results from FWA model are still in good agreement with the experimental data even as the advance ratio becomes smaller. Strong rollup of the wake at low

The FWA wake model is also partially validated with the experimental measurement of the wake flow of propeller DTMB 4119 by Jessup [

On the evaluation of KT and KQ, the viscous effects are simplified with an empirical viscous pitch correction [

Correlation of the FWA model with the latest experimental data can also be very helpful to improve the methods described in this paper. Typically the PIV measurement of the wake flow of the INSEAN E779A propeller by Di Felice et al. [

Support for this research was provided by the U.S. Office of Naval Research (Contract no. N00014-07-1-0616 and N0014-10-1-0931) and Phases V and VI of the “