On the working principle of pre-swirl stators and on their application benefit and design targets

Abstract

Background:

In an early design stage of ship and propeller or in a propeller retrofit case one may consider a pre-swirl stator (PSS) to recover rotational energy.

Objectives:

A model for the working principle of a PSS was to be developed and then to be applied for power saving estimates. This model should also support the PSS design procedure.

Methods:

The PSS ‘working principle model’ is based on a very fundamental quantity related to a propellers characteristic, namely the circulation distribution. An extended model addresses hub vortex contraction losses. In specific cases a BEM (Boundary Element Method) based propeller analysis is performed. To confirm actual PSS designs, a RANS/BEM coupling method is invoked.

Results:

Global parameters as $K_{T}$ and CTH rule the rotational losses of a propeller and determine the energy recovery potential of a PSS accordingly. The potential of a PSS is considerably enlarged, if it works properly against the hub vortex. Estimates for realistic power savings due to a PSS installation are 4–5% when the hub vortex suppression is included.

Conclusions:

The simple PSS ‘working principle model’ delivers very reasonable predictions. RANS results and available full scale trial measurements can be reproduced using this approach.

Keywords

Energy saving device propulsion retrofit propeller hub vortex propeller rotational losses

Nomenclature

Propeller disk area ( $π R^{2}$ )

CTH

Thrust Load Coefficient, $CTH = T / (1 / 2 ρ A u^{2})$

Propeller diameter

Propeller radius

Propeller thrust

Δ T

Added thrust requirement of propeller in combination with an ideal pre-swirl stator

Δ T_{S}

Added thrust requirement of propeller if combined with a real pre-swirl stator

Undisturbed axial inflow under open water

Δ u

Added axial velocity at propeller disk due to actuator disk model

V_{s}

Ship speed

\overline{v_{t}} (r)

Circumferential mean of the tangential velocity behind the propeller

\overline{v_{t}^{'}} (r^{'})

Normalized $\overline{v_{t}} (r)$ with normalized radius as argument ( $\overline{v_{t}^{'}} (r^{'}) = \overline{v_{t}} (r) / (Ω R)$ )

w_{m}

Mean axial velocity defect, so that $(1 - w_{m}) V_{s}$ stands for the mean axial flow entering a device

Γ (r)

Circumferential mean of circulation distribution for one blade

Γ^{'} (r^{'})

Normalized circumferential mean of one blade circulation $= Γ (r^{'}) / (Ω R^{2})$

Γ_{o}^{'}

Amplitude defining a standard circulation distribution so that $Γ^{'} (r^{'}) = Γ_{o}^{'} f (r^{'})$

f (r^{'})

Shape function

K_{T}

Propeller trust coefficient $= T / (ρ n^{2} D^{4})$

Γ_{H}^{'}

Normalized circulation of a propeller blade at the hub $= Γ_{H} / (Ω R^{2})$

Number of blades

r_{H}^{'}

Normalized hub radius

r^{'}

Normalized radial position within 2D vortex structure $r_{C}^{'}$ Non-dimensional radius of a cavitating vortex

p_{o}

Environmental pressure

p_{v}

Vapor pressure of water

p^{'}

normalized pressure $= p / (1 / 2 ϱ {(Ω R)}^{2})$

Δ p^{'}

Normalized drop from surrounding pressure $= (p_{o} - p) / (1 / 2 ϱ {(Ω R)}^{2})$

ρ_{w}

Density of water

σ_{n}

Tip speed based cavitation number $= (p_{o} - p_{v}) / (1 / 2 ϱ {(Ω R)}^{2})$

Angular Frequency of propeller $= 2 π n$

1. Introduction

The stand-alone propeller is introducing swirl to the fluid. This swirl assembles angular momentum, which is closely linked to the torque at the propeller. In view of propulsion efficiency the swirl gathers rotational energy but it is not supporting the process of thrust generation. In other words, this portion of energy is pure losses and does not contribute to the propulsion of the ship.

The general idea to prevent swirl losses and improve propulsion efficiency is actually not new and among a lot of other sources one can find discussion of devices introduced to recover rotational energy in [8] and [6]. From the pure hydrodynamic point of view a post-swirl device may represent the simpler choice. An optimum finite bladed stator for the post position was already considered by Betz [1] who introduced the simplification, that the swirl is generated by a single central vortex. However, an efficient post-swirl device is probably difficult to mount and to maintain behind a propeller. A Pre-Swirl Stator (PSS) in contrast will experience lower velocity magnitudes from the incoming flow. Under the assumption of similar span and comparable lift and drag, the wetted surfaces can be larger for the PSS fins. Though in the above we used a strength and fatigue argument in favour of the pre-swirl stator solution, strength and fatigue aspects are not addressed in detail here.

We concentrate on the hydrodynamic aspects of the stator/propeller interaction.

The present paper starts with a general classification and quantification of effects, which reduce propeller efficiency. It then concentrates on the deficits due to swirl production and approaches the structure of the rotational flow behind the propeller. An additional negative force is linked to the hub vortex and causes further deficits in propeller efficiency. This hub vortex effect is described by a supplementary loss model.

Then we come to the task of optimising a PSS solution. We introduce a generic model for the stator/propeller interaction, which identifies the increased propeller thrust as a decisive effect. Searching for a design target, we define parameters within the generic model to simulate the presence of a real stator and a real propeller. Reasonable parameter settings give a first link to the optimum pre-swirl device for given propeller characteristics. The general findings at this point emphasize the ‘reduced effort’ design approach, meaning that one should intentionally create reduced swirl compared to what is shed at the propeller.

The generic interaction model may be extended and become sensitive to the actual main data of a propeller. An adequate procedure is introduced and definite figures for the RPM shift at the propeller are obtained. Having stressed the benefits of a PSS the treatment of the interaction process thus reveals a disadvantage of this recovery solution. In case of a stator retrofit with intentionally untouched propeller the swirl added to the propeller inflow will drop the propeller RPM.

It is also demonstrated in this paper, how the stator geometry can be detailed for a specific hull and a genuine propeller. Accordingly one creates an entry for a PSS design process.

A favourable stator performance and a strong influence of the PSS on the hub vortex development were identified in the European FP7 project GRIP. In GRIP the ‘Uljanik’ test case was set up and intensively studied up to full scale trials with mounted pre-swirl stator. The paper finally demonstrates how the RANS calculations and trial results fit to the simple model for the early assessment of power saving potentials.

2. Assessment of pre-swirl stator benefits

2.1. Applying a model for propeller specific losses

To be specific in analysing the propeller energy losses, one has to account for the actual propeller geometry and perform an analysis, which should not only deliver the overall (full scale) propeller efficiency but also a decomposition of the energy losses. The actual consumed power would be decomposed into the useful power appended by losses of different classes, such as axial, rotational, non-uniformity and viscous losses. Here we applied a Vortex Lattice Method (VLM, see for example [4] and [5]) with additional analysis output. A Boundary Element Method (BEM, see for example [2]) could be applied alternatively.

Fig. 1.

Potential theory propeller model (left, here a Vortex Lattice Method) and derived propeller specific losses (right). On the right the gap between actual propeller open water efficiency (here full scale assumed) and 1.0 is divided into different contributions. This gap assembles losses due to axial velocity increase (ax_loss), viscous effects (vis_loss), slipstream rotation (ro_loss) and non-uniformity (non-uni_loss), the latter resulting indirectly from the constant velocity increase assumed in the axial loss model.

A typical decomposition of losses is shown in Fig. 1. As the potential theory models usually do not capture the hub vortex contraction process along the boss cap, the corresponding added rotational losses are excluded in Fig. 1. A simple model for such supplementary losses will be introduced later in this section. Here we considered it reasonable to treat the rotational losses first up to a level valid for an infinite cylindrical and frictionless hub.

In Fig. 1 we assumed open water conditions for a propeller originally designed for in-behind operation and realized a typical thrust coefficient $K_{T}$ . Using the propeller thrust load coefficient CTH we can derive the ideal actuator disk efficiency: $\begin{matrix} (1) & η_{i} = 2 / (1 + \sqrt{1 + C_{T H}}) \end{matrix}$ The axial losses are then given by the ‘offset’ between 1 and $η_{i}$ of the actuator disk, which would neither show rotational, viscous nor non-uniformity losses. The rotational losses included in Fig. 1 are here defined as the rate of shed kinetic energy, which is related to the circumferential mean $\overline{v_{t}} (r)$ of the tangential velocity behind the propeller. The structure of $\overline{v_{t}} (r)$ will be discussed more in detail in the following section. Further losses are viscous losses, available since the calculations are based on an inviscid approach with supplementary viscous corrections.

The losses due to non-uniformity are filling the gap caused by the idealized treatment of axial and rotational losses. They are supposed to assemble mainly axial losses, the former not covered by the simple actuator model as it introduces a homogeneous added axial velocity $Δ u$ at the propeller disk. Accordingly non-homogeneities in radial and circumferential direction are suppressed. Similar defects hold for the rotational loss model, which was intentionally based on a circumferential mean tangential flow $\overline{v_{t}} (r)$ behind the propeller. For the case displayed in Fig. 1 the rate of energy transmitted into $\overline{v_{t}} (r)$ amounts to 3.3% when based on the total power at the propeller. The following section will explain how this figure has been derived.

Compared to the open water situation, an increased and case-dependent non-uniformity loss level characterizes the in-behind case.

2.2. A generic model for the swirl structure behind the propeller excluding hub vortex contraction

As the above model for the specific losses at a propeller of given geometry under given operational conditions is not yet sensitive to capture hub cap details, we also set up the generic rotational loss models in a decomposed manner. First we introduce a model valid for an infinite cylindrical and frictionless hub. Consequently any plane for the integration of rotational energy related to the circumferential mean value $\overline{v_{t}} (r)$ shows the hub radius as inner boundary (see Fig. 2).

Fig. 2.

Model for the system of singularities and boundaries (infinite hub) ruling the circumferential mean value $\overline{v_{t}^{'}} (r)$ . The trailing vortex systems of the propeller blades can be simplified to non-helical structures when addressing this mean value.

When we gave an example for the ranking of losses with Fig. 1, it was also emphasized, that this ranking depends on the operational conditions and on the propeller geometry. In an early design stage it may be sufficient and intended to use global propeller performance data like thrust, torque and rate of revolutions, keeping in mind that in an initial stage it may introduce too much effort to work with detailed propeller geometries. Stepping over from a specific propeller to a generic substitute we proceeded as follows:

Using a normalized amplitude $Γ_{o}^{'}$ to define a standard normalized circulation distribution $Γ^{'} (r^{'})$ so that $Γ^{'} (r^{'}) = Γ_{o}^{'} f (r^{'})$ with $f (r^{'})$ a shape function, we can come via Stokes’ theorem to a normalized swirl $\begin{matrix} (2) & \overline{v_{t}^{'}} (r^{'}) = \overline{v_{t}} (r) / (Ω R) = Z Γ_{o}^{'} f (r^{'}) / (2 π r^{'}) . \end{matrix}$ In the above Ω and Z represent the angular frequency of the propeller and number of blades respectively. The emphasized shape function $f (r^{'})$ is given on the left of Fig. 3. The hub circulation of the blades enters as a decisive part of the shed vortex system $d Γ (r) / d r$ , the latter giving quite straightforward $\overline{v_{t}} (r)$ (after inserting $1 / r$ ). The shape of the normalized swirl velocity $\overline{v_{t}^{'}} (r)$ can be taken from the middle of Fig. 3.

Fig. 3.

Using a unique circulation shape function ( $f (r^{'})$ , on the left) one can access the losses due to slipstream rotation w/o hub vortex contraction for basically any propeller (on the right). The profile of the tangential velocity in the slipstream ( $\overline{v_{t}^{'}} (r)$ , middle) is an interims result ( $r^{'} = r / R$ ).

Considering a tube of length $Δ x$ the kinetic energy of the rotating flow would amount to $\begin{matrix} (3) & Δ E_{rot} = 2 π Δ x Ω^{2} R^{4} \int_{r_{H}^{'}}^{1} \frac{1}{2} ρ_{w} {(\overline{v_{t}^{'}} (r^{'}))}^{2} r^{'} d r^{'} . \end{matrix}$ Inserting (2) into (3), one arrives at $\begin{matrix} (4) & Δ E_{rot} = \frac{1}{4 π} ρ_{w} Δ x Γ_{o}^{^{'} 2} Z^{2} Ω^{2} R^{4} \int_{r_{H}^{'}}^{1} f {(r^{'})}^{2} / r^{'} d r^{'} . \end{matrix}$ Due to this formula the rotational losses are ruled (in quadratic form) by $Z Γ_{o}^{'}$ . Without giving more details we remark, that $Z Γ_{o}^{'}$ (now in linear form) rules as well the thrust T via the approximation $\begin{matrix} (5) & T = ρ_{w} Ω^{2} R^{4} Z Γ_{o}^{'} \int_{r^{'}_i}^{1} f (r^{'}) r^{'} d r^{'} . \end{matrix}$ Accordingly a linear entry of $Z Γ_{o}^{'}$ remains for the energy ratio $Δ E_{rot} / (T Δ x)$ . When the thrust as approximated in (5) is normalized to give the thrust coefficient $K_{T}$ , it can be recognised that the thrust coefficient $K_{T}$ is directly proportional to the energy ratio $Δ E_{rot} / (T Δ x)$ .

Finally the flux of rotational energy across the propeller disc is needed to set up a power ratio. The energy flux through the disk would read $Δ E_{rot} / Δ t$ and this introduces via $Δ x / Δ t$ the axial component at the disk, namely $u + Δ u$ (linked to the thrust load coefficient CTH), as the second relevant parameter for the rotational loss level. Typical settings for $Z Γ_{o}^{'}$ (equivalent to characteristic settings for $K_{T}$ ) and $(u + Δ u) / u$ (corresponding to characteristic choices for CTH) give values $Δ E_{rot} / Δ t$ of 3–4% (related to the power at the propeller). Besides the shape function $f (r^{'})$ a generic hub radius ratio of 0.2 was assumed here.

Consider there would be a device introduced in front of the propeller for nothing else but for the production of counter-swirl by redirecting the straight inflow. It would show a negative force $- Δ T = Δ E_{rot} / Δ x$ and accordingly a propeller working together with such a device would have to increase thrust to maintain the ‘system thrust’. This interaction effect is addressed in more detail further below. At the propeller itself an additional negative force is introduced by the hub vortex. To quantify this force and to investigate, how it can be avoided, we need a supplementary model, which is introduced in the next section.

2.3. The hub vortex model

Losses due to rotation will tend to rise if one includes a model for the vortex tightening process occurring along the boss cap. The boss cap will experience a strong suction force while this process is supposed not to influence the axial force at the propeller blades. The confined vortex structure leaving the hub will lead to the low pressures acting on the boss cap. As a consequence the level of losses given in Fig. 1 might slightly change when this effect is built-in.

When the propeller hub vortex is taken into account, another clear benefit for producing pre-swirl compared to working on post-swirl can be assured apart from the strength arguments. If this vortex has developed already, useful energy may be hard to recover from such a confined and strong structure. On the other hand the hub vortex development can be suppressed conveniently by a pre-swirl device.

As outlined below, the thrust diminishing effect of this process may be estimated via two significant parameters. Introducing first the normalized hub circulation $Γ_{H}^{'}$ , the normalized pressure drop $Δ p^{'}$ as a function of the distance $r^{'}$ form the hub vortex center can be expressed as: $\begin{matrix} (6) & Δ p^{'} (r^{'}) = \frac{Z^{2} Γ_{H}^{' 2}}{(4 π^{2})} (\frac{1}{r^{^{'} 2}}) \end{matrix}$ Assuming that this pressure field is acting on the boss cap we need a relevant radius to start the pressure integration. The phenomenon of hub vortex cavitation is supporting the target of a simple model for added losses due to hub vortex development. Provided that the pressure drops exactly down to vapour pressure, i.e. $Δ p^{'} = σ_{n}$ , the radius for the cavitation filled core of the vortex $r_{c}^{'}$ may be derived as $\begin{matrix} (7) & r_{c}^{'} = (Z Γ_{H}^{'}) / (2 π \sqrt{σ_{n}}) . \end{matrix}$ With $r_{c}^{'}$ as lower boundary for the pressure integration and under the assumption, that the pressure around a 2D vortex with strength $Z Γ_{H}^{'}$ is acting on the hub cap, all relevant settings are assembled. A normalized force $F_{x}^{'}$ can be evaluated by integration from $r_{c}^{'}$ to $r_{H}^{'}$ as follows from the pressure acting on the projection of the hub cap: $\begin{matrix} (8) & F_{x}^{'} = F_{x} / (\frac{1}{2} ϱ Ω^{2} R^{4}) = 2 π \int_{r_{c}^{'}}^{r_{H}^{'}} Δ p^{'} (r^{'} d r^{'}) \end{matrix}$ Substituting (6) into (8): $\begin{matrix} (9) & F_{x}^{'} = \frac{Z^{2} Γ_{H}^{' 2}}{(2 π)} \int_{r_{c}^{'}}^{r_{H}^{'}} \frac{1}{r^{'}} d r^{'} = \frac{Z^{2} Γ_{H}^{' 2}}{(2 π)} (ln r_{c}^{'} - ln r_{H}^{'}) \end{matrix}$ For a further evaluation of above formula one may introduce empirical assumptions: $Z^{2} Γ_{H}^{' 2}$ is assumed to give typically a value of 0.01 whereas ( $ln r_{c}^{'} - ln r_{H}^{'}$ ) is assumed to read typically 1.8, this would result in a normalized axial force $F_{x}^{'}$ equal to 0.0028. To make $F_{x}^{'}$ comparable with the propeller thrust, we define the normalised thrust $K_{T}^{'}$ in a similar way as $F_{x}^{'}$ : $\begin{matrix} (10) & K_{T}^{'} = T / (1 / 2 ϱ Ω^{2} R^{4}) \end{matrix}$ $K_{T}^{'}$ introduced as above is related to the conventional thrust coefficient $K_{T}$ by $\begin{matrix} (11) & K_{T}^{'} = K_{T} \cdot 8 / π^{2} \end{matrix}$ Assuming that a characteristic $K_{T}$ would read 0.175, one obtains $K_{T}^{'}$ equals to 0.14. Finally by relating $F_{x}^{'}$ and $K_{T}^{'}$ we obtain the ratio of $F_{x}^{'} / K_{T}^{'}$ to be 2%.

This would mean that the suppression of the hub vortex contraction process would lead to 2% less thrust to be provided by the propeller blades, which – considering typical propeller efficiencies – would roughly correspond to a 3% saving in power!

Since 2D pressure fields have been assumed in the simplified hub vortex contraction model, a certain over-estimation on the suction force may be introduced here. At this stage of the hub vortex loss model and in view of a real stator hardware showing certain losses itself, it seems reasonable to work with an added 1.5% regain to cover savings due to hub vortex elimination.

3. Basic results on the relation of propeller and pre-swirl stator

3.1. The spread of the rotational loss level for a single propeller

Propellers differ in size and are tailored to work at specific combinations of thrust, RPM and inflow speed. The question arises, whether a wide spread of rotational losses is to be expected. Assuming, that the shape of the circulation over radius $f (r^{'})$ is unchanged, the relevant coefficients for the level of rotational losses are propeller thrust coefficient $K_{T}$ and propeller thrust load coefficient CTH, as stated already in the section above. The thrust coefficient is closely linked to the circulation amplitude $Γ_{o}^{'}$ ruling the energy $Δ E$ inside the tube of length $Δ x$ . Instead of considering CTH one may refer to the advance coefficient J as the alternative parameter, necessary to specify the volume flux into the tube of length $Δ x$ .

Figure 3 presents the propeller specific rotational loss level (excluding hub vortex contraction losses) as a function of J and $K_{T}$ . Assuming, that propeller designs drop in $K_{T}$ when J decreases, it appears that nearly any propeller arrives at a rotational loss level between 3–4%, without consideration of the hub vortex contraction losses.

The profile of the tangential velocity in the slipstream – as displayed in the middle of Fig. 3 in normalized form – is an interim result. The curve’s shape is determined by the function $f (r^{'})$ . Normalizing as in Fig. 3 with the incoming axial velocity u, the magnitude is determined by $K_{T}$ and J.

It is to be pointed out however, that the generic shape function $f (r^{'})$ is somehow reflecting a conservative view as far as the shedding of rotational energy is concerned. Therefore, in the next subsection, variations of $f (r^{'})$ are considered.

3.2. Comparing rotational loss levels for open water and in-behind

As the circulation developing on a given propeller is also depending on the inflow conditions, one should investigate the difference of rotational losses in open water and for the in-behind environment. Using a propeller designed for the in-behind case, we recognised a clear wake effect on the level of rotational losses.

Fig. 4.

The radial distributions of the normalized local axial flow $1 - w (r^{'})$ (left, normalized by mean axial velocity $1 - w_{m}$ ); the related circulation distributions on the propeller blade (middle, ‘ $Γ_{J 045}$ ’ points out that the reference speed based advance coefficient remained the same, namely 0.45); the quantification of related rotational losses (right).

Figure 4 demonstrates that the circulation distribution (here valid for one blade and normalized by $Ω R^{2}$ ) further concentrates to the hub when changing the wake slope. The rotational losses increase when the hub region is further loaded. With $Z = 4$ for this case the non-dimensional circulation met at the hub (see Fig. 4, middle) confirms the setting $Z^{2} Γ_{H}^{' 2}$ to read typically 0.01. The hub vortex contraction model was however not further adapted when summarizing the losses on the right of Fig. 4. For losses due to the hub vortex development we added here globally 1.5%. This is in line with the earlier (possibly conservative) estimates based on our hub vortex model.

As a conclusion one may emphasize to detail the distribution of load on the propeller and improve the preliminary results on rotational losses, which were based on a generic circulation.

3.3. On thrust requirement under pre-swirl stator presence and on a concept for the optimum ‘stator- effort’

The structure and the magnitude of the rotational losses have been addressed in the above. Next we need to consider an efficient process to prevent or convert them. If one decides in general for a pre-swirl device, at first sight it seems reasonable to aim at the profile of the tangential flow as predicted behind the propeller (see Fig. 3) and just reverse the direction. It was one demand in the GRIP project to derive realistic power gains for a stator implementation, i.e. account for ‘side-effects’ coming along with a physical pre-swirl stator installation.

To estimate the potential benefit of a pre-swirl device, we introduce first a stator/propeller interaction scenario valid for an ideal stator. In the first step, we keep also the propeller geometry unspecified. Fixing the two parameters that rule the rotational losses at the propeller, namely J and $K_{T}$ , the model delivers additional valuable information, such as the relative thrust increase $Δ T / T$ of the propeller. For an ideal stator, this thrust increases has just compensated the force $- Δ T$ acting on the stator itself. As already stated above, for such a device $- Δ T$ is identical to its shed rotational energy per unit length $Δ E_{rot} / Δ x \cdot Δ T / T$ grows as the tangential velocity profile given in the middle of Fig. 3 is enhanced.

Matching the profile of the propeller to 100% might not represent the optimum for a physical stator installation since there are always losses present in the swirl generation process. One may introduce $f_{vt}$ (usually $< 1$ ) as the ratio of swirl actually created by the ‘physical’ stator to the amount of swirl shed by the stand-alone propeller. In combination with $f_{vt}$ two related parameters may be fixed before one starts a variation of $f_{vt}$ aiming at the optimum performance of a real stator. First an efficiency $η_{S} < 1$ can downgrade the ideal stator ( $η_{S} = 0.8$ is emphasized currently without further knowledge on the stator geometry). Introducing $η_{S}$ and letting $Δ T_{S}$ denote the axial force on a real stator, the power invested to produce swirl would read $Δ T_{S} (1 - w_{m}) V_{s}$ while $η_{S} T_{S} (1 - w_{m}) V_{s}$ would stand for the useful (i.e. circumferential uniform) pre-swirl.

Similarly one can introduce with $η_{Z}$ a ‘pre-swirl acceptance’ efficiency of the real propeller due to its finite blade number. Setting either these parameters to unity and alternatively to the emphasized settings $(0.8, 0.8)$ one can compare the corresponding power reductions at the propeller.

Fig. 5.

Introducing efficiencies ( $η_{S}$ and $η_{Z}$ ) for the pre-swirl generation process emphasizes a reduction of pre-swirl effort $f_{vt}$ for real stator hardware to achieve optimum recovery. The grey curve holding for $η_{S} = η_{Z} = 1$ shows the optimum recovery point at 100% pre-swirl effort. The minimum ratio of Power with ( $P_{with}$ ) and without stator (P) is shifted to 70% pre-swirl effort (dark curve), when $η_{S}$ and $η_{Z}$ are set to realistic values (here 0.8, 0.8).

This comparison results into the two curves as displayed in Fig. 5. The usage of the emphasized settings ( $η_{S} = η_{Z} = 0.8$ ) will ascertain smaller but realistic power reductions at the propeller. With these settings for $η_{S}$ and $η_{Z}$ a ‘swirl effort’-parameter $f_{vt}$ of 0.7 gives the maximum power saving.

3.4. Effects of a specific stator design on the operation of a specific propeller

Stepping from the early assessment of the power reduction effect of a pre-swirl stator to the behaviour of a specific propeller it is reasonable to detail the distribution of the swirl $\overline{v_{t}^{'}} (r^{'})$ . A concrete distribution under actual wake conditions may be derived from a lifting line code or especially a BEM analysis. The higher thrust requirement of the propeller has already been identified as an unavoidable effect of pre-swirl generation. Solely the stator causes and determines this higher thrust. From the propeller side it is important to determine the new RPM and the new power.

Though addressing here RPM and power at a specific propeller we still assume that the generic character of the stator remains. This generic device is given the ability to generate the specific $\overline{v_{t}^{'}} (r^{'})$ in shape, not necessarily in magnitude since the ‘stator efficiency’ $η_{S} < 1$ tends to diminish the optimum level for the incoming swirl. The remaining parameter $η_{Z}$ is dropped since the response of the propeller on the distribution $\overline{v_{t}^{'}} (r^{'})$ (usually diminished by $f_{vt}$ ) is not estimated but calculated. To quantify the new propeller RPM, the thrust increase $Δ T_{S}$ is used to fix the working point for the propeller under pre-swirl inflow.

The change of the propeller operation with pre-swirl stator is illustrated in Fig. 6. The $K_{T}$ -curve valid for operation under swirl shows a significant shift. A substantial RPM drop is found in this illustration case (corresponding to an increase of the J-value where the propulsion point is met). The higher required thrust diminishes the RPM drop only slightly.

Fig. 6.

Typical shift of propeller operation point with and without pre-swirl stator. The curves labeled ‘Thrust_w/o’ and ‘Thrust_with’ identify the thrust requirement when thrust is normalized as $K_{T}$ and plotted over the advance ratio J. $J_{with}$ and $J_{w / o}$ identify the actual operation points of the propeller without and with stator.

Linked to Fig. 6 the values for normalized thrust and power without and with stator can be taken from Table 1. Here we also give a comparison of the actual power savings and the predictions from the simple rotational loss model in case of full recovery ( $η_{Z} = 1.0$ ). A comparison of realized and ideal savings (columns 4 and 5) would lead to $η_{Z} = 0.9$ in this case (3.1% against 3.4%). Additional recovery prospects due to hub vortex suppression (typically 1.5%) are not covered with Table 1.

Table 1

Advance coefficient J, thrust coefficient KT, normalized absolute thrust and power without as well as with stator according to BEM analysis for a specific propeller case

		w/o stator	with stator
1	J	0.698	0.747
2	KT	0.191	0.230
3	Absolute Thrust (KT/J²)	0.392	0.412 (5.2% higher)
4	Realized absolute Powers (10KQ/J³)	0.966	0.936 (3.1% savings)
5	Ideal absolute Powers (10KQ/J³)	0.966	0.933 (3.4% savings)

4. Pre-swirl stator installation for a given hull

4.1. Design targets

One has to account for the inflow to the stator as experienced behind the hull when stepping from the idealized stator to a real device (as sketched in Fig. 7). The stator sketched in Fig. 7 is however hardly suitable for a single propeller case. Behind a hull the single propeller will usually work in an inclined flow. For the layout of a pre-swirl device the inclined flow situation introduces an asymmetry. The propeller already partly receives properly directed incoming swirl (for a right turning propeller on Stb) and partly experiences just the opposite (on Port). Depending on the angle of inclination it seems natural to concentrate stator fins on Port rather than to distribute them uniquely.

Fig. 7.

A ‘real’ stator device showing cambered sections and a reasonable outline (here as a 5-bladed version with symmetrical arrangement of fins).

If the flow information valid at the stator plane was obtained via a ‘bare hull’ RANSE analysis, this flow should be corrected for propeller-induced velocities. The former findings based on the generic PSS model and depending on the efficiency factor settings ( $η_{S} = η_{Z} = 0.8$ ) suggested a relative effort $f_{vt} = 70$ % to be spent on pre-swirl generation. The analysis of the ‘Uljanik’ case (see further below) leads to a somewhat higher $f_{vt}$ -setting (85%), necessary to verify the actual power savings. Due to this practical experience it is currently emphasized to start a stator design process with the assumption, that the best performance point corresponds to $f_{vt} = 0.75$ .

Thus the $f_{vt}$ -setting governs the proposed design procedure. Following the direct analogy between overall circulation and overall swirl, the PSS fins should in total establish a circulation equal to the mean propeller circulation (all blades) diminished by $f_{vt}$ . If especially the total circulations of fins and blades meet at the root sections, one can in addition eliminate hub vortex losses to a large extent. The sections of the stator fins should work at shock free entrance (stagnation point at LE) and consequently the fins are to be twisted individually. Even the outline of fins may be a matter of distinct adjustment. A moderate camber to chord ratio is to be demanded, so that the chord remains as the relevant parameter to meet the shape of the ( $f_{vt}$ -reduced) propeller circulation.

The analogy of circulation and swirl production leads to the comparison given in Fig. 8. On the left of this figure the circulations on stator fin and propeller blade are similar in magnitude and assumed reversed in sign. The contributions to the mean swirl by one fin and one blade are thus the same in magnitude but they cancel each other due to the opposite directions. As we assume a 3 bladed stator and a 4 bladed propeller in Fig. 8, the overall stator-‘effort’ for swirl production reads 75% of the total swirl shed by the propeller. This corresponds to the realization of $f_{vt} = 0.75$ , which, together with the matching of circulations in shape, one can call a ‘design target’ for the pre-swirl stator hardware. The corresponding design constraint demands, that the surface pressure on the fins sections should be smooth (as shown on the left of Fig. 8).

Fig. 8.

If $f_{vt}$ was set to 0.75 each fin of a 3-bladed stator should show a circulation distribution as one blade of a 4-bladed propeller (left, plotted over span). The pressure coefficient on the fin’s sections should show shock free entrance (right).

Most of the considerations given above assume uniformity in tangential direction for the propeller inflow and for the propeller loading, i.e. the propeller blade position $φ_{o}$ causes no changes. As mentioned above, the usual case of a propeller behind an ordinary mono-hull already violates this assumption. As a consequence one has to review the design targets sketched in Fig. 7 and Fig. 8 for most of the cases met in practice. It seems natural to rely on the idea of submitting a circumferential uniform swirl to the propeller, which means an asymmetry in swirl production effort in cases where the initial tangential inflow component is significant. Especially the symmetrical arrangement of fins shown in Fig. 7 may be unfavourable then. The circumferential uniformity of the total tangential flow to the propeller may be addressed as the 2nd design target for a stator installation, which aims at reducing the non-uniformity losses.

There might be additional recovery prospects, if these ideal propeller inflow conditions expressed with the 2nd design target are realized. It appears that additional research is necessary to forecast such added gains depending on the actual constellation.

4.2. The ‘Uljanik’ test case

In the FP7 EU project GRIP the task was given to design an energy saving device (ESD) for a full scale validation ship. The test case ship available in this project was a 182 m Bulk Carrier built by the partner Uljanik Shipyard. HSVA designed an ESD in form of a pre-swirl stator for this ship. This was also the device finally selected in the project and installed on the Uljanik bulker. The trials were very successful (see reference [3]) and showed a power saving of 6.8% due to the pre-swirl stator, which is higher than the numerical (RANS) predictions performed before the trial (see reference [7]). On the basis of HSVA’s viscous calculations we will below investigate, what the simple estimation models introduced above would predict if supported by some interim results from the RANS analysis.

The code that is mainly used by HSVA, FreSCo+, solves the incompressible, unsteady Navier-Stokes-equations (RANSE). For the Uljanik bulker case it was applied for full scale conditions. The propulsion situation was modelled via a RANS-BEM coupling, internally labelled as the ‘RANS-QCM’ coupling. Via this approach, one obtains the wake components at the propeller plane including the effects of streamline contraction, the so called ‘effective’ wake. As the full scale analysis was done at HSVA under propeller operation with as well as without stator, there were two different profiles available for the effective tangential wake component, which is giving the level of pre-swirl arriving at the propeller. The contour plots for these tangential components ( $v_{t}^{'}$ ) are given in Fig. 9, where the ship speed was used for normalization of velocities and the propeller radius R served to normalize lengths. Positive values for $v_{t}^{'}$ identify a direction running with a right turning propeller. The view onto the propeller plane is from behind. As can be seen, the tangential velocities in a great portion of Port side have the positive sign, which means a light running of the propeller in this region.

Fig. 9.

‘Uljanik’ case: effective tangential wake w/o stator (top) and effective tangential wake with mounted pre-swirl stator (bottom) according to full scale hull flow analysis using FreSCo+. The tangential component is normalized by ship speed.

Fig. 10.

‘Uljanik’ case: Difference of effective tangential wakes with and w/o pre-swirl stator (based on ship speed).

From the bottom plot of Fig. 9 (giving the contours for the tangential component with mounted stator) one may identify a reasonably rotational symmetric pattern, which has been established by the pre-swirl device. To link Fig. 9 to the simple models formerly described in this paper, we subtracted the top plot from the bottom plot. The result is shown in Fig. 10, which illustrates clearly, that the highest pre-swirl effort was done on Port (where indeed all three stator fins were positioned in a further upstream plane). As the models presented above are based on the circumferential means of the tangential flow, such mean values were derived from the pattern of Fig. 10 and are presented in Fig. 11.

Fig. 11.

Actual pre-swirl profile (v_t(w-w/o)) and ‘matching’ profile (dashed), the latter from the simple estimation model described in this paper with effort parameter set to f_vt = 0.85.

This figure also includes a ‘matching’ distribution, which was obtained from the generic circulation of the simple model. The matching curve involves the actual values of $K_{T}$ and CTH holding for the Uljanik propeller. In addition, to achieve this quite perfect agreement of curves in Fig. 11, the ‘pre-swirl effort’ parameter $f_{vt}$ was set to 0.85. As could be deduced from Fig. 5, the ‘standard’ setting of the efficiency parameters for $η_{S}$ and $η_{Z}$ would not really emphasize such a high value in view of power input and power ‘payback’. It seems however reasonable to review these parameters for situations where the propeller is operating in a substantially inclined flow. Deriving the settings for the propeller efficiency parameter $η_{Z}$ expressing ‘swirl acceptance’, it was assumed, that the propeller was meeting initially zero tangential flow everywhere.

With inclined flow as present for in-behind conditions one is in a different situation: the initial propeller inflow can be regarded as circumferential non-uniform swirl and – with the help of the PSS – an overall more uniform swirl is lastly provided to the propeller. For the propeller, experiencing not only swirl but also an enhanced uniformity, one might even consider that 1.0 doesn’t represent the upper limit for the parameter $η_{Z}$ . The latter idea arises from the view, that any measure compensating the unfavorable initial tangential flow pattern (caused by the inclined inflow) gives an additional ‘payback’ due to a smoother propeller performance.

Fig. 12.

Using the simple estimation model to locate the ‘Uljanik’ pre-swirl stator layout in the general effort(f_vt)/payback(P_new)-diagram.

If we set $η_{S} = 0.8$ and $η_{Z} = 1.0$ to treat the Uljanik case with the simple model, we arrive at the recovery prospects as function of the swirl effort $f_{vt}$ . This function is shown in Fig. 12 (dashed curve). The circumferential means of $\overline{v_{t}^{'}} (r^{'})$ from RANS calculations and from the simple model agree best when setting $f_{vt} = 0.85$ (Fig. 11). Having thus identified $f_{vt} = 0.85$ as entry-parameter to locate the Uljanik case in our simple model and re-setting $η_{Z}$ as above, we arrive at 4.5% power reduction as shown in Fig. 12. In addition we have to fix the hub vortex contraction losses. If we assign further 2.0% power benefits (thus increasing our previously introduced 1.5% by 0.5% as a kind of fitting process) to a hub vortex elimination attained by the stator, we end up with overall 6.5% savings. With the aid of a refitted $η_{Z}$ and a less conservative treatment of the hub vortex we reproduce the full scale trial results (up to 6.8% power gain) quite well.

5. Summary and conclusion

In the above we presented simple methods to quantify the rotational losses of a ‘stand alone’ propeller in general. The plural ‘losses’ is justified since for convenience we divided the process into two parts. Part one excludes the further contraction of the shed vortex system after passing the propeller blades while part two treats the added loss effect due to the hub vortex contraction.

Introducing in the radial direction a generic distribution for the blade load, the losses caused by these two processes are deduced from the global parameters $K_{T}$ and CTH (excluding contraction) and the overall circulation leaving at the hub $Z Γ_{H}$ (when the contraction effect is added) respectively.

It was further outlined how estimations on power reduction valid for a mounted pre-swirl stator can be deduced from the idealized treatment of rotational losses. To link a theoretically optimum pre-swirl stator to a realistic stator design a ‘reduced effort’ philosophy was introduced. Discussing on the RPM drop under pre-swirl presence, it was considered unavoidable to reference the actual propeller geometry at this stage.

Lastly we leave the theoretical assumptions on the flow environment of propeller and stator gradually and formulate design targets for real stator hardware. An in-house Boundary Element Method has been used at the Hamburg Ship Model Basin to link design targets and stator geometry. It guarantees a quick investigation of various stator alternatives. In viscous flow computations involving the hull, the stator and the propeller the final fin design should be proven and confirmed (and probably altered accordingly). This step was very decisive for the Uljanik test case, since the RANS analysis enables the final selection of the optimum fin design (see reference [7]).

For the Uljanik tests case (intensively studied in the GRIP project up to full scale trials with mounted pre-swirl stator) we demonstrated how the RANS calculations and trial results fit to the simple model for the early assessment of power saving potentials.

As a conclusion it may be stated that the simple estimation model proved to deliver very reasonable results. Due to the typically inclined flow situation of a propeller behind a mono hull the simple modelling may require adjustments to cover additional stator benefits arising from an improved homogeneity of the flow to the propeller. Also the quantification of the hub vortex contraction losses is complex and has not been completed yet (especially one should try a RANS modelling with cavitation involved). Currently it appears, that the globally derived 1.5% power savings for a ‘counter hub vortex’ device may be too conservative.

Footnotes

Acknowledgement

The authors really acknowledge the supports obtained via the European FP 7 project GRIP. This project allowed for both, financial funding and exchange and cooperation with other participants. These aspects appeared to be essential for the realization of the above study on pre-swirl devices.

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