## Abstract

Three-dimensional particle image velocimetry (PIV) experiments were conducted in the immediate near wake and up to seven diameters downstream of a three-bladed marine propeller model operating in two different inflow conditions: one with imposed freestream turbulence with intensity of 7% and streamwise integral length scale comparable to propeller geometry, and the second experiment with a quiescent inflow conditions as a reference. The resulting Reynolds number based on propeller chord and relative velocity is $Re0.7R$ = 4.7 × 105. All components of radial transport of mean flow kinetic energy are analyzed and the largest contributor to the fluxes is found to be correlated to Reynolds shear stresses, resulting in radially outward flux in the wake. Two regions of the near wake are distinguishable with downstream extent dependent on the level of external turbulence. In the first region, immediately behind the propeller, shed tip vortices are very coherent and undergo grouping and roll-up around each other and the second region where the vortex merger process is complete and characterized by breakdown of vortices into small-scale turbulence. The latter region was found to occur earlier in the experiment with external turbulence. Conditional statistics of velocity fluctuations were employed and they show that outward interactions and sweep events contribute the most to the transfer of mean flow kinetic energy from the inner wake to the freestream.

## 1 Introduction

Turbulent wakes of rotor systems such as marine propellers, helicopter blades, and flow energy extractors such as wind and marine current turbines are flows of significant practical importance. Wakes developing behind these devices are fundamentally different in comparison to the wakes of bluff bodies, particularly in the near wake region, due to the presence of the helical structure of vortex filaments that are continuously shed at the rotor's blade tip and root. The evolution of individual rotor wakes from the standpoint of instability modes of the helical tip vortex structure has received considerable attention in the literature through analytical, numerical, and experimental results as summarized recently by Ref. [1]. Early analytical work by Widnall [2] defined three different stability modes of near wake structure breakdown and all were experimentally confirmed for marine propellers by Felli et al. [3]. They showed that the mutual inductance stability mode has the greatest impact on the onset of near wake instability and subsequent breakdown of helical structures. Mutual inductance mode is described as a process where in a row of closely spaced helical vortices, if they are spaced closely enough and with small perturbations, neighboring vortices will start to group and roll up around each other resulting in a vortex merger and subsequent break down into small-scale turbulent structures. Felli et al. [3] have shown that the inception of this process is dependent on the number of propeller blades as well as the blade loading condition, specified by the nondimensional advance coefficient J (see definition later in text).

Lignarolo et al. [4] described in detail the roll-up process in the near wake of a single two-bladed wind turbine at two different turbine-loading conditions. Of particular interest to the present discussion, Lignarolo et al. [4] also addressed an important question regarding mean flow kinetic energy entrainment in the wake region and the role the mutual induction instability mechanism has on the kinetic energy transport and the associated downstream wake re-energization. This is of significant practical importance for wind turbines and, recently, for marine current turbines from the standpoint of design and optimization of turbine farm layouts. Newman et al. [5] provides a good summary of the body of work related to the mutual interaction of individual turbine wakes in wind farms. They also experimentally studied various terms in the mean flow kinetic energy equation to quantify vertical transport of mean flow kinetic energy and address the role external turbulence, in their case turbulent boundary layer, has on the re-energization of the individual wakes in the turbine array. In the marine propulsor community, apart from the studies concerning propellers operating in the wake of a ship's hull whose primary focus is on propeller performance, cavitation, and propeller efficiency [6,7], the body of knowledge on how upstream flow conditions affect the near wake dynamics is limited. Recent work by Felli and Falchi [8] showed that the dynamics of the propeller near wake vortical system in oblique inflow conditions, that is, propeller operating in an inclined condition (finite yaw angle), has different behavior with respect to the axisymmetric conditions and that this affects the mutual interaction instability mechanism.

The present experimental study examines the spatial evolution of the marine propeller's near wake in the early stages from the immediate wake behind the propeller up to seven propeller diameters downstream for two inflow conditions: one with imposed external turbulence of specific intensity and integral scale, and the second one with clean inflow conditions as a reference. All components of the vertical fluxes of mean flow kinetic energy are calculated from available particle image velocimetry (PIV) data and compared for the two inflow conditions. The influence of external turbulence on the process of mutual induction is examined and it was found that external turbulence enhances the vortex roll-up mechanism resulting in the earlier breakdown of individual vortices into small-scale turbulence. Conditional sampling is performed to analyze the mechanisms of mean flow kinetic energy transport in the outer portions of the wake and how this transport is influenced by inflow (external) turbulence.

## 2 Experimental Setup

Experiments were conducted in the U.S. Naval Academy's recirculating water tunnel shown in Fig. 1 that has 0.41 × 0.41 m2 cross section and 1.8 m long test section. The tunnel features a 4.6:1 contraction section at the upstream entrance into the test section. For the present set of experiments (see Figs. 1(b) and 2), propeller model was supported by a vertical rectangular stainless steel bar (12.7 × 6.4 mm2) rigidly attached to the propeller shaft bearing housing. Additionally, a streamlined strut (chord length of 45 mm and a maximum thickness of 9 mm) was attached over the steel bar to prevent any flow-induced vibration of the propeller support structure. The propeller shaft was connected to a 19 mm diameter flexible shaft that ran upstream through the tunnel contraction section, through the turbulence-generating grid and through the penetration on the tunnel wall to a coupling connected to the computer-controlled motor that sets propeller rotational speed. Flexible shaft is shrouded by a 35 mm diameter flexible casing (see Fig. 2) and 220 mm long fairing is used as a transition from shaft diameter to the propeller hub diameter to reduce shaft wake signature.

Fig. 1
Fig. 1
Fig. 2
Fig. 2

Two sets of stereoscopic PIV (SPIV) measurements were performed in the propeller model wake: one with passive turbulence generating grid placed upstream from the propeller, and the second experiment in the quiescent inflow condition, that is, clean tunnel condition without turbulence generating grid. For the latter experiment, levels of freestream turbulence in the test section were not measured in this experiment but previous extensive measurements by Refs. [9] and [10] in the same facility and clean tunnel condition have shown inflow turbulence intensity does not exceed 0.5%.

### 2.1 Propeller Model.

The propeller model is a left-handed, fixed pitch, three-bladed propeller with diameter D = 129.5 mm, hub diameter of 15 mm, pitch/D ratio of 1.03 at r/R = 0.7, and expanded blade area ratio of 0.643 (Fig. 2). Performance characteristics of this propeller model have been obtained through a standard marine propeller open water test conducted in the USNA Hydro Mechanics laboratory 37 m (120 ft) towing tank in quiescent inflow conditions, and are presented in Fig. 3. Here, thrust coefficient $KT=(T/ρn2D4)$, torque coefficient $KQ=(Q/ρn2D5)$, and propeller open water efficiency $η=(KT/KQ)(J/2π)$ are plotted versus the advance coefficient $J=(U∞/nD)$. T and Q are propeller developed thrust and torque, ρ is the water density in the tank, n is the propeller rotational speed in revolutions per second (rps), and $U∞$ is the speed of advance of the propeller, which in the case of the open water test is the tow carriage speed. Uncertainty analyses applicable to propeller open water tests were conducted as per Ref. [11] at three different advance coefficients and are shown as error bars in Fig. 3. Levels of uncertainty at 95% in KT range from ±0.007 to ±0.009, and in torque coefficient, KQ ±0.0025 and are approximately constant.

Fig. 3
Fig. 3

The advance coefficient of J =0.6 was chosen as the operating point for the present near wake measurements. This corresponds to propeller rotational speed n =27 rps, nominal tunnel velocity of $U∞$ = 2 m/s resulting in Reynolds number based on diameter of $ReD=2.5×105$, and Reynolds number based on propeller chord length at 0.7R and relative velocity $Urel=U∞2+(0.7RΩ)2,Re0.7R=4.7×105$. Since no forces measurements were made during the wake study, conservation of linear momentum analysis was performed using available SPIV data as described below to estimate propeller developed thrust during the present experiments, and the results are shown in Fig. 3 by two values of averaged thrust coefficients KT = 0.283 ± 0.017 at J =0.59 and KT = 0.295 ± 0.035 at J =0.56 for the experiment with external turbulence and quiescent case, respectively. Slight difference in the advance coefficient stems from the fact that $U∞$ differs between the two experiments by 5%. Uncertainty in KT values is estimated as twice the standard deviation [12] of distribution of thrust coefficient over the first two diameters behind the propeller model.

Additionally, an in-house designed shaft position indexing system featuring a Hall Effect sensor is mounted to the propeller hub to phase lock the PIV image pair capture with the reference blade position. Considering that 27 pulses per second, which correspond to the propeller rotational speed, is larger than the PIV laser repetition rate (7.5 Hz), additional electronic firmware was designed to provide laser triggering (TTL pulse) every 10th passage of the reference blade. This effectively determined the phase-locked PIV acquisition rate to 2.7 Hz. For the free runs, the PIV acquisition rate was at the system maximum of 7.5 Hz in camera straddle mode.

### 2.2 Turbulence Generating Grid.

For the experiment with external turbulence, the turbulence-generating grid shown in Fig. 2 was placed nominally 3.5 propeller diameters (D) upstream from the propeller plane. It is a uniform square bar grid with a mesh width of 38.6 mm, a horizontal/vertical bar thickness of 9.7 mm, and a resulting solidity of 44%, similar to the conventional grid used by Krogstad and Davidson [13]. The grid pattern differs from the published design by a circular hole with a diameter of 43 mm in the middle of the grid to accept a shaft connected to the propeller model. The grid was designed to control two parameters of the imposed external turbulence: the streamwise integral length scale of the turbulence, L11, and the streamwise turbulence intensity. The former is defined in Pope [14], and the latter is defined as, $Ix=(u′2¯/U∞)$, where $u′2¯$ is the root-mean-square of velocity fluctuations. Prior to the present experiments, detailed measurements of L11 and Ix were performed in uniform flow in the water tunnel about 0.5D from the tunnel centerline, using two-dimensional PIV with the turbulence grid in place but without propeller model installed in the tunnel. Resulting decaying turbulence has turbulence intensity of Ix = 7% at the propeller plane and decaying to Ix = 5% by the end of the water tunnel test section. The realized turbulence integral scale is L11 = 0.2D at the propeller plane.

### 2.3 Stereoscopic Particle Image Velocimetry.

Three-dimensional PIV measurements were made in the vertical plane containing the propeller axis of rotation at the recirculating channel centerline using two TSI 8MP (3320 × 2496 pix2) cameras in stereoscopic configuration. Cameras were placed at approximately 45 deg angle with respect to the incoming flow. To minimize optical aberrations, two acrylic (8 mm thick walls) prisms filled with water were attached to the tunnel windows as recommended in Ref. [15], ensuring that each camera's optical path is perpendicular to the window surfaces. Flow field illumination is provided through the bottom acrylic window of the tunnel by Evergreen Nd-YAG laser (532 nm) with an average output of 200 mJ. The seeding particles used are Potter Industries SH400S20 with 13 μm diameter and specific gravity of 1.1. The entire setup is mounted on a traversing system (see Fig. 1(a)) allowing translation over the entire test section of the water tunnel. Spatial coverage in the wake was accomplished via “tiling” of 15 individual (0.18 × 0.09 m2) fields of view (fov) covering distance from x = 30 mm downstream from the propeller plane up to approximately x = 7D, and radially about ±0.6D from the propeller centerline (y =0). Two rows of five fov positions covered top ($y/D>0)$ and bottom ($y/D<0)$ part of the wake, with 15% overlap radially, and an additional five fov centered at y = 0. The interrogation window for vector analysis was 32 × 32 pixels2 with a 50% overlap resulting in a spatial resolution of 0.18 mm taken as twice the interrogation window. At each spatial location, two separate measurements were made: phase-locked, that is, PIV image pairs were triggered at a specific blade location, and unconditioned free runs. The average uncertainty in instantaneous velocity vectors was obtained from TSI insight4 g software and is based on Ref. [16] method, $ϵu¯$ = ±0.09 m/s (0.16 pixels) at 68% confidence level for both cameras. This uncertainty encompasses random and systematic errors typically found in a PIV experiment and will be propagated to a 95% confidence level in calculating uncertainties of other measured quantities derived from the PIV data.

In the analysis that follows, we utilize Reynolds and Hussain [17] triple velocity decomposition defined as:
$ui(t)=Ui+uĩ(t+nτ)+u(t)′$
(1)
where $ui(t)$ is instantaneous velocity, t indicates time, Ui is the time-averaged mean velocity, and $ui(t)′$ are random turbulent fluctuations. Term $uĩ(t+nτ)$ is used to separate contributions of period velocity fluctuations from the random turbulent fluctuations. Hence, $uĩ$ represent velocity fluctuations associated with coherent vortices shed from the propeller blades, τ is the time period between two successive reference blade passages, and n here represents the number of reference blade passages. Index i = 1, 2, or 3 is used for the present three-dimensional measurements indicating streamwise u, vertical v, and lateral w velocity components, respectively, as indicated in Fig. 2. Two different measurements were obtained at each spatial location: phase-locked with reference propeller blade in vertical position ($ϕ=0$) and unconditioned free runs. With phase-locked datasets, averaging operation is performed as follows:
$〈ui(x,y)〉ϕ=0=1Nf∑k=1Nfui(x,y,tk)ϕ=0$
(2)
where Nf  =  600 is the number of realizations of $u(x,y,t)ϕ=0$ at a given phase. Random turbulent fluctuations are then calculated as
$ui′(x,y,t)ϕ=0=ui(x,y,t)ϕ=0−〈ui(x,y)〉ϕ=0$
(3)
It should be noted that $ui′$ random fluctuations as defined in Eq. (3) are pertinent to a single phase. In the remainder of the text, phase averaging at a single phase $ϕ=0$ is implied. With free runs, time-averaged velocity is calculated as
$Ui(x,y)=1N∑k=1Nui(x,y,tk)t$
(4)
where N =600 is the number of PIV realizations in each unconditioned dataset. Velocity fluctuations for unconditioned dataset are calculated using Reynolds decomposition $u′i(x,y,t)t=ui(x,y,t)t−Ui(x,y)$ and are used in estimating uncertainty in time-averaged velocity from the expression [18]
$ϵU=ut′2¯+εu2¯Nt95,N−1$
(5)

where overbar indicates time averaging, $t95,N−1$ is the value of Student's t-distribution at 95% confidence level, and N−1 is the number of independent samples.

## 3 Results

### 3.1 Mean Velocity Profiles.

Figure 4 shows the downstream distribution of time-averaged velocities scaled by $U∞$ for two experimental cases. Streamwise velocities, U(x, y) appear axisymmetric with two maximums occurring approximately at y locations corresponding to the propeller blade radial positions developing maximum thrust and a reduction in velocity magnitude in the region ($−0.1) due to presence of hub wake. Lack of symmetry in U(x, y) at the wake outer edges and in the hub wake particularly visible in Fig. 4(b) are attributed to the slight difference in top and bottom inflow velocities due to presence of the propeller strut (see Fig. 2), tip vortex pairing at the outer edges of the wake, and hub vortex meandering in and out of the vertical PIV measuring plane. These points are further discussed later in the text. Magnitudes of V(x, y) velocity are small except in the hub wake region where they are consistently negative. One would expect the region of negative vertical velocity above the propeller axis and positive vertical velocity below the propeller axis. Thus, it is likely that PIV measuring plane was slightly offset in the z-direction. Out of plane mean velocities W(x, y) are positive above and negative below propeller centerline consistent with the left-handed propeller rotation.

Fig. 4
Fig. 4

The overall effect imposed external turbulence has on the mean velocities is to “stabilize” wake at large scales. Meandering or undulation of the wake outer edges and the hub wake is clearly visible in the case without external turbulence (see Figs. 4(b), 4(d), and 4(f) at $x/D≈3$ and $x/D≈5$). Felli et al. [3] measurements have shown that the propeller hub vortex experiences instabilities that depend on both the number of blades and loading conditions. For their three-bladed propeller model, hub vortex radial oscillation amplitude at advance coefficient J =0.6 in a tunnel with freestream turbulence intensity of less than 2% was found to be 0.11D. For the present experiment without external turbulence hub vortex oscillation amplitude in radial direction estimated from the distribution of phase averaged out of plane velocity $〈w〉$, not shown here for brevity, is found to be the same, and 0.08D for the case with external turbulence. At the outer wake edges, spatial undulations can be observed at $x/D≈3$ and $x/D≈6$, and are not present for the case with external turbulence.

In order to elucidate further downstream evolution of time-averaged velocity profiles, Tennekes and Lumley [19] self-similar wake velocity scaling is used. Although this scaling is not appropriate for the developing near wake, it is found to be instructive in the analysis that follows. For this scaling, locally defined velocity and length scale are needed. For a propeller system that inputs energy to the flow, streamwise wake surplus velocity US is defined as
$US(x,y)=U(x,y)−U∞$
(6)

The wake velocity scale is defined as a maximum of $US(x,y)$ velocity profile $max(US(x=const,y))=US MAX(x)$ for a given downstream location. The corresponding local length scale, wake half-width δ is defined as the radial location at which the surplus velocity $US(x,y)$ is equal to $12US MAX(x)$.

Figure 5 shows the surplus velocity $US(x,y)$ normalized by $US MAX(x)$ as a function of normalized radial distance $y/δ$ for a range of downstream x/D locations. Only five profiles for the experiment with external turbulence are shown for visualization purposes. Uncertainty estimates using Eq. (5) are shown with vertical error bars at four different locations in the wake. Uncertainty spatially varies across the wake with the largest magnitude of ±0.053 m/s occurring in the region of maximum velocity shear approximately corresponding to $y/δ$ = 1. Velocity profiles clearly do not collapse due to the developing wake. With increasing distance from the propeller plane, locations of the peak velocities move inward toward the centerline, while in the region $1 velocity profiles adjust to satisfy conservation of mass resulting in downstream wake widening there. Axisymmetry of the profiles is clearly evident, except at the wake outer edge $y/δ=1.5$ where profiles at x/D =0.7 and 2 show $≈$4% increase in surplus velocity not explained by uncertainty there, in comparison to the opposite radial location at $y/δ=−1.5$. This is attributed to the additional blockage due to propeller strut (see Fig. 2) and is consistent with the 3% area ratio of the frontal strut area to the half of the tunnel cross-sectional area.

Fig. 5
Fig. 5

Before proceeding further with the analysis, and for the completeness considering that during experiments no force measurements were performed, control volume approach of conservation of linear momentum in conjunction with available streamwise time-averaged mean velocity profiles from PIV data was used to estimate propeller thrust as

$T=2πρ∫0y=1.5δU(x,y)US(x,y)ydy$ employing axisymmetry assumption and radial integration up to $y=1.5δ$. Magnitudes of calculated thrust coefficients KT for two experimental cases are shown in Fig. 3. KT for the case with external turbulence differs from the open water value by 10%. Discrepancy, although small, is statistically significant and is attributed to the difference in Reynolds number between the open water tests at $Re0.7R=2.9×105$ versus present water tunnel experiment at $Re0.7R=4.7×105$. Magnitude of KT for the quiescent case is slightly larger than the value for the case with external turbulence implying larger developed thrust, as it should be due to slightly lower advance coefficient, and also has a larger uncertainty that is attributed to the spatial variability in thrust coefficient within wake's first two diameters. Uncertainty levels for the quiescent case suggest that open water tests conducted in the towing tank and the momentum analysis from water tunnel data have the similar magnitudes at a 95% confidence level.

In the wake region shown in Fig. 5 encompassing $−0.75 all surplus velocity profiles exhibit a velocity deficit with a characteristic secondary minimum, $min(US(x))$, occurring approximately at the propeller centerline $y/δ$ = 0 and decreasing with downstream distance from the propeller plane. Local velocity deficit magnitude defined as: $[US MAX(x)−min(US(x))]/U∞$ is further examined and its downstream distribution is presented in Fig. 6 for two experimental cases. We interpret this velocity deficit as the manifestation of the presence of the hub and blade root vortices as well as the propeller lower momentum input due to the reduction of blade relative angle of attack. Because of the PIV fov arrangement, analysis is performed separately for the top $y/D>0$ row, bottom $y/D<0$ row, and center row of fov, resulting in two independent estimates of velocity deficit at each downstream position. By the end of the wake portion covered by the measurements ($x/D=7$), the velocity deficit in both cases converges to approximately 15% of $U∞$ indicating that propeller hub wake persists a long downstream distance, an observation consistent with Felli and Falchi [8] results. However, the velocity deficit evolves differently between two experimental cases. In the case with external turbulence velocity, deficits have smaller magnitudes and are consistent between different experimental runs. On the other hand, velocity deficits in the case without external turbulence are larger in magnitudes and differ between experimental runs suggesting hub vortex oscillations both vertically and laterally. As a fixed vertical PIV measuring plane is cutting different sections of the oscillating hub vortex, the magnitudes of both $US MAX(x)$ and $min(US(x))$ velocities change. Also, due to a lack of convergence between different experimental runs, it is possible that 600 PIV realizations were not sufficient to obtain converged time-averaged mean statistics in $US MAX(x)$ and $min(US(x))$ at the early stages of the wake due to intermittency of the oscillating hub vortex for the quiescent experimental case. Raising the level of external turbulence to 7% results in a spatially more consistent velocity deficit as well as an earlier transition ($x/D≈2.5$) toward a linearly decreasing trend in comparison to transition in the quiescent case, which occurs further downstream at ($x/D≈3.5$) without a clear downward trend.

Fig. 6
Fig. 6

### 3.2 Mean Flow Kinetic Energy.

The transport equation for the mean flow kinetic energy $(1/2)UiUi$, neglecting viscous effects and using triple velocity decomposition defined in Eq. (1) for a single-phase [4,17] is given as
$(Uj+uj̃)∂∂xj(12UiUi)=−1ρUi∂∂xi(P+p̃)−Ui(∂uĩ∂t+Uj∂uĩ∂xj)−∂∂xj(Ui〈ui′uj′〉+Ui〈uĩuj̃〉)−(−〈ui′uj′〉−〈uĩuj̃〉)∂Ui∂xj$
(7)

Here, the first term on the right-hand side represents transport by mean pressure P and coherent pressure fluctuations $p̃$ and the second term is the convection by the mean flow of coherent velocity fluctuations. The third term is the transport of mean flow kinetic energy by phase-averaged Reynolds stresses $〈ui′uj′〉$ and coherent velocity fluctuations $〈uĩuj̃〉$, and the fourth term is the production of turbulence by mean shear working against Reynolds stresses and its coherent counterpart $−〈uĩuj̃〉$. In the analysis that follows, the focus is on quantifying the terms within the gradient of the third term on the right-hand side of Eq. (7) in vertical y-direction perpendicular to the incoming flow and propeller axis. These terms represent vertical fluxes of mean flow kinetic energy across horizontal surfaces and represent energy exchange between wake and the bulk flow [4,5]. In Cartesian coordinates, these terms are: $−(U〈u′v′〉+V〈v′v′〉+W〈w′v′〉)$ and $−(U〈ũṽ〉+V〈ṽṽ〉+W〈w̃ṽ〉)$, and are all measured in the present experiments. The largest uncertainties of the phase-averaged stresses over the wake domain covered by the present experiments are ±0.017 m2/s2 (18%) for shear stress and ±0.03 m2/s2 (6%) for normal stresses at 95% confidence level. For the coherent Reynolds stresses, $〈uĩuj̃〉$ uncertainties are ±0.13 m2/s2 (12%) and ±0.33 m2/s2 (11%) and rapidly diminish downstream from the propeller. These calculations are based on uncertainty propagation method described in Ref. [18].

In order to elucidate the behavior of these terms, conditional sampling of Reynolds stresses based on quadrant analysis is performed [4,20]. For example, conditional sampling for the phase-averaged shear stress $〈u′v′〉$ is defined as
(8)

where Nk is the number of occurrences of combination of $u′l$ and $v′l$ fluctuations in a quadrant K. Four quadrants are defined: quadrant 1 (outward interaction) with $u′>0; v′>0$, quadrant 2 (ejection) with $u′<0; v′>0$, quadrant 3 (inward interaction with $u′<0 ;v′<0$, and quadrant 4 (sweep) with $u′>0; v′<0$.

Figures 7 and 8 show isocontours of conditionally sampled $〈u′v′〉$ for cases with and without external turbulence. Ejection and sweep events are premultiplied by –1 to show all events at the same scale. The largest shear stresses occur at tip vortices and in the hub wake clearly identifiable in all quadrants due to spatial “locking” by phase averaging. In all quadrants shown in Fig. 7 starting at $x/D≈1.5$ a group of three vortices can be seen undergoing the merging process by rolling-up around each other. The fact that a group of three vortices is interacting is consistent with a three-bladed propeller. By $x/D=4$ merging process is completed after which individual vortices break down into smaller-scale turbulent structures. In comparison, the group of three neighboring vortices shown in Fig. 8 for the case with external turbulence are shown initially interacting at the same spatial location, $x/D≈1.5$, and are smeared beyond recognition by $x/D=3$ without undergoing any of the intermediate features such as roll-up visible in Fig. 7. Thus, it appears that external turbulence disrupts the mutual induction instability process resulting in an earlier transition to turbulent structures responsible for the further downstream development of the propeller wake as discussed next.

Fig. 7
Fig. 7
Fig. 8
Fig. 8

Black dots shown in Figs. 7 and 8 represent radial locations where $y=±0.75δ$. These radial locations approximately correspond to the peaks of the mean velocity profiles (see Fig. 5) and are interpreted here as a proxy for boundary separating outer wake from the wake core containing hub vortex. In Fig. 8 downstream from $x/D>3$, outward interaction (quadrant 1) and sweep (quadrant 4) events have the largest magnitudes occurring in the outer wake. Sweep events are traditionally interpreted in boundary layer flows as bringing high-momentum fluid from more energetic freestream flow downward into the slower moving fluid. Here, sweep events can be interpreted as taking high-momentum fluid from the wake core toward the outside slower-moving freestream. Similarly, outward interaction (quadrant 1) can be interpreted as taking the high-momentum fluid from the wake core and pushing it outward toward the freestream. Ejection (quadrant 2) and inward interaction (quadrant 3) events, on the other hand, have their respective largest magnitudes occurring approximately at the wake boundary indicated by black dots. Ejection events bring high-momentum fluid from the region where maximum in velocity occurs $y/δ=±0.75$ and bring it inward toward the hub wake region. Similarly, inward interactions are pushing high-momentum fluid inward toward the centerline on the top side of the propeller. We conjecture here that ejection and inward interactions events are responsible for the sudden change in slope of deficit velocity seen in Fig. 6 at $x/D≈2.5$.

Interpretations just described are more evident for the case with external turbulence than for the case without. This is because the spatial extent of individual tip vortices and the process of mutual induction followed by breakdown into turbulence are longer for the case without external turbulence as discussed above. As shown in Fig. 7, past $x/D>4$ outward interaction (quadrant 1) and sweep (quadrant 4) events have the largest magnitudes occurring in the outer wake. This is further discussed later in this section.

Figure 9 shows the isocontours of normalized unconditioned vertical flux of mean kinetic energy by Reynolds shear stresses, $−U〈u′v′〉$ for the two experimental cases. In the top part of the wake $−U〈u′v′〉$ is negative, suggesting dominance of the outward and inward interactions resulting in upward transport of mean flow kinetic energy, whereas in the bottom part of the wake, this term is positive suggesting dominance of the ejection and sweep events and downward transport of mean flow kinetic energy. As will be shown next, the magnitudes of two other terms in the vertical flux of kinetic energy $−V〈v′v′〉$ and $−W〈w′v′〉$ are an order of magnitude smaller and their isocontours are not shown for brevity. Phase-averaged vertical fluxes are further quantified by calculating downstream distribution of spatial averages as
$QUi〈u′iu′j〉(x)=−1A U∞3∫x−Δx2x+Δx2∫y=0.75δy=ymaxUi〈u′iu′j〉dy dx$
(9)
where integration is performed horizontally over a wake section of $Δx=0.08D$ in the streamwise direction and vertically from $y/δ=±0.75$ up to the edges of the PIV field of view. Thus, magnitudes of $QUi〈u′iu′j〉(x)$ represent normalized spatially averaged fluxes in the outer portion of the wake. Spatial variability of fluxes within each integration area A is quantified by calculating uncertainty at 95% confidence level [12] as
$εUi〈u′iu′j〉=1U∞3 t95,n−1std(−Ui〈u′iu′j〉)n−1$
(10)

where n is the number of samples of $−Ui〈u′iu′j〉$ in each wake region covered by the integral and std is the standard deviation of fluxes within integral bounds. The downstream distributions of unconditioned $QU〈u′v′〉, QV〈v′v′〉$ and $QW〈w′v′〉$ are presented in Fig. 10 for the two experimental cases as a cumulative sums together with respective cumulative envelopes of uncertainty.

Fig. 9
Fig. 9
Fig. 10
Fig. 10

All contributors to the vertical mean kinetic energy flux except $QV〈v′v′〉$ are consistent, that is, in the top part of the wake they are negative, indicating upward flux of energy, and at the bottom they are positive, indicating downward flux. One would expect axisymmetry in flux magnitudes between top and bottom parts of the wake, but that is not the case as upward (negative) fluxes are consistently smaller in the top part of the wake. Plausible explanation is that vertical fluxes at the upper part of the wake ($y/D>0$) are affected by the presence of the propeller strut and consequently slightly larger blockage as discussed in Sec. 3.1. $QV〈v′v′〉$ term is positive everywhere indicating downward flux over the entire outer portion of the wake. This is presumably due to small but negative mean vertical velocity as shown in Fig. 4(d). The largest contributor to the fluxes is clearly $QU〈u′v′〉$ as its magnitude is approximately larger by an order of magnitude than other terms. To the best of the authors' knowledge, this is the first time that mean flow kinetic energy flux terms are reported for the marine propeller wake. In the near wake of wind turbines, it has been shown that $−U〈u′v′〉$ flux is also the dominant term in the vertical transport of mean flow kinetic energy [5,20].

Figure 10 also highlights the difference between the downstream evolution of vertical fluxes for two experimental cases. In the case without external turbulence (Fig. 10(b)), the magnitude of normalized vertical flux $QU〈u′v′〉$ at the location closest to the propeller is 1 × $10−3$ and increases gradually to a magnitude of 13 × $10−3$ by $x/D=5$. For the case with external turbulence, (Fig. 10(a)) magnitude of $QU〈u′v′〉$ also increases gradually from 5 × $10−3$ and reaches almost constant magnitude of 18 × $10−3$ by $x/D=3$. Thus, even though magnitudes of $QU〈u′v′〉$ are larger throughout the near wake for the case with external turbulence, the relative increase between two cases is approximately the same. This result implies that the effect of external turbulence is to redistribute differently available kinetic energy imparted to the fluid by the propeller action.

The same calculations are performed on vertical fluxes by phase-averaged coherent velocity fluctuations $〈ũiũj〉$ and are shown in Fig. 11. Results appear noisier in comparison to the fluxes by Reynolds stresses suggesting larger spatial variability within the wake integration regions. The key feature is that vertical fluxes by the coherent fluctuations are comparable in magnitude to the random fluxes in the immediate near wake, but diminish rapidly downstream. This decrease in cumulative sum from the magnitude immediately behind the propeller implies a change in sign of vertical fluxes. In the case with external turbulence, the net contribution from all coherent vertical fluxes diminishes to zero by $x/D≈2.5$. In this region, however, the cumulative sum of $QU〈ũṽ〉$ (Fig. 11(a)) has large uncertainty and any interpretation of changes from positive to negative values are not statistically significant to make any conclusions. The same can be said for $QU〈ũṽ〉$ for the case without external turbulence shown in Fig. 11(b), except that fluxes diminish to nearly zero by $x/D=4$. For both experimental cases, locations of diminishing contribution of vertical fluxes by coherent fluctuations coincide with the spatial locations in the wake where individual tip vortices are no longer discernible (see Figs. 7, 8, and 9) and represent locations where mutual induction mechanism of vortex pairing and roll-up is complete. Qualitatively similar results have been reported by Lignarolo et al. [4] for the wind turbine operating in quiescent inflow conditions with an exception that the location of diminishing coherent vertical fluxes coincided with the spatial location of tip vortex breakdown for a two-blade rotor system.

Fig. 11
Fig. 11

Having established that Reynolds shear stresses are the major contributor to the vertical fluxes of mean flow kinetic energy, and that coherent velocity fluctuations do not play a role in the vertical flux past the location where tip vortices breakdown, of interest is to examine the mechanism of energy transport in the outer wake. To this end, Fig. 12 shows the cumulative sum of conditionally sampled $QU〈u′v′〉$ given by Eq. (8) and grouped as follows: contributions by inward interactions and ejections, $−U(〈u′v′〉3+〈u′v′〉2)$ (red symbols) and contributions by outward interactions and sweep events $−U(〈u′v′〉1+〈u′v′〉4)$ (black symbols). Figure 12(a) shows that in the bottom part of the wake with downward net positive flux, sweep and ejection events are consistently larger than outward/inward interaction events. Similarly, in the top part of the wake with upward net negative flux, outward and inward interaction events are dominant. However, past $x/D=2.5$ sweep and outward interactions become the primary mechanism for outflow of mean flow kinetic energy across the wake outer boundary and keep increasing downstream, whereas ejection and inward interaction events cease to contribute to the cumulative sum as indicated by flat horizontal trend. Exactly the same interpretation can be made for the case without external turbulence, except that separation of ejection/inward interaction events occurs at approximately $x/D=4$.

Fig. 12
Fig. 12

## 4 Conclusions

In this study, near wake development of the marine propeller was examined experimentally using SPIV for two inflow conditions: one with a clean steady inflow and the second one with passive grid generated turbulence with 7% intensity and integral scale of 0.2D comparable to propeller geometry. It was found that imposed external turbulence stabilizes the near wake both in the inner and outer portions of the wake. In the inner wake portion, external turbulence reduces the large-scale oscillations (meandering) of the hub vortex by 27% evidenced in both the time-averaged mean velocity profiles and in hub vortex deficit velocity. However, as present results show, external turbulence does not affect the downstream extent of the hub vortex, as manifested by velocity deficit at the propeller centerline found to be 15% of $U∞$, which has been shown by Felli and Falchi [8] to extend well beyond the present experiment of seven propeller diameters downstream.

In the outer part of the wake, whose boundary is defined as the radial locations $y>±0.75δ$, where δ is wake half-width, external turbulence disrupts the downstream extent of tip vortices merging process, which is established to be the most relevant instability mechanism of rotor systems near wakes [3,4]. As a result, transition to turbulence structures responsible for the downstream wake development occurs earlier in the wake at $x/D=2.5$ in comparison to $x/D=4.5$ for the case without external turbulence. The onset of the instability process was not analyzed in this work, but the experimental evidence presented in Fig. 7 qualitatively show the early stages of the mutual induction occurring at the same x/D locations for both experimental cases. Parametric study by Felli et al. [3] predicts the onset of tip vortex mutual induction instability process for three-bladed propeller in quiescent inflow conditions and advance coefficient J =0.6 at approximately one-dimensional downstream from the propeller.

Cumulative distribution of spatially averaged vertical fluxes $QUi〈u′iu′j〉$ indicate upward transport at the top and downward transport at the bottom of the wake, thus top/bottom outflow of mean flow kinetic energy from the wake and into the less energetic freestream. It was found that vertical fluxes are primarily correlated to the phase averaged Reynolds shear stresses as term $QU〈u′v′〉$ has an order of magnitude larger contributions than $QV〈v′v′〉$ and $QW〈w′v′〉$ consistent with wake studies on wind turbine rotor systems [4,5]. Similarly, all contribution to vertical fluxes by coherent velocity fluctuations was found to be negligible past the location where tip vortices breakdown into smaller structure.

The mechanism of vertical energy flux revealed through conditional sampling indicates that relative contributions to positive downward flux by ejection/sweep events and negative upward flux by inward/outward interaction events depends on the location in the wake corresponding to the downstream location where tip vortex breakdown is completed. For the case with external turbulence past $x/D=2.5$ and in the case without external turbulence past $x/D=4$, sweep and outward interactions become the primary mechanism for outflow of mean flow kinetic energy and keep increasing downstream to the full extent of the present experiment ($x/D=7$). Ejection and inward interaction events appear to be more relevant in the early wake stages leading up to the downstream location where tip vortices breakdown occurs and have no significant contributions to the fluxes further downstream.

Overall, the earlier transition to turbulence structures in the outer wake due to the presence of external turbulence results in different redistribution of the available mean flow kinetic energy imparted by the propeller to the fluid. An earlier allowance for entrainment of kinetic energy from the wake to the surrounding fluid will impact the development of the far wake and possibly result in an earlier disintegration of the wake and reduction of propeller wake signature.

## Acknowledgment

The authors would like to thank the staff of the U.S. Naval Academy Hydrodynamic Laboratory for their contributions in designing and building the experimental setup as well as assistance in performing propeller open water tests. We would also like to thank the Midshipmen Trident Scholar program at the U.S. Naval Academy for the support of Bennitt Hermsen work.

## Funding Data

• Office of Naval Research (Grant No. N0001418WX00060; Funder ID: 10.13039/100000006).

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