Improving model scale propeller performance prediction using the k − k L − ω transition model in OpenFOAM

Abstract

The effect of laminar to turbulent flow transition plays an important role for the prediction of model scale performance, which is of utmost interest for the development of scaling approaches entirely based on Computational Fluid Dynamics calculations. The recent inclusion of transition models (either based on local correlations, like the $γ - {Re}_{θ}$ , or on the concept of kinetic laminar energy, like the $k - k_{L} - ω$ ) in many RANS codes fosters their application for improving the model scale prediction of propeller performance. In the present work the numerical results using the well-established SST $k - ω$ and the $k - k_{L} - ω$ turbulence models available in OpenFOAM are presented and compared with towing tank experiments for three test case propellers. The influence of turbulence parameters (i.e. turbulence intensity and turbulent viscosity ratio at inlet) is discussed, at first for the ERCOFTAC T3A flat plate validation case, through which useful guidelines for propeller performance predictions using transition sensitive turbulence models are derived. By using these relationships, a significant improvement of numerical predictions of propeller forces is achieved, with discrepancies with respect to model scale measurements appreciably reduced if compared to usual fully turbulent calculations. At the same time the limitations of the adopted transitional model are discussed based on the systematic analyses carried out for three test cases.

Keywords

Propeller performance laminar-turbulent transition RANS OpenFOAM

1. Introduction

The prediction of ship powering and performance is based on the knowledge of propeller open water characteristics, which are usually determined by model scale towing tank tests. The dimensions of the experimental facility, however, pose serious limitations to the functioning conditions as well as to the sizes of the models, leading to significant scale effects. Differences in the flow regimes are due to very different functioning Reynolds numbers and then the full scale characteristics are not similar to model scale measurements. Most of the full scale predictions of propeller performance are base on relatively simple extrapolation methods of the model scale measurements but the increasing demand of high performance propellers requires more accurate prediction tools for full scale performance, in order to simultaneously comply with interactions of the propeller with the full scale ship wake [20] and account for side effects like cavitation inception and pressure pulses which could be reasonably influenced by scale effects. Over the years, different scaling methods have been proposed. The first attempt to collect empirical approaches dates back to early ’50 during the 6th ITTC conference [29]. Roughly, scaling approaches can be divided into statistical methods, methods based on equivalent profile/propeller, strip methods and, recently, CFD calculations. The most widely applied statistical method is the ITTC’78 scaling approach [30] [31] proposed by the 15th ITTC committee. Among the methods based on the equivalent propeller analogy, probably that proposed by [47] is the best known, while among strip methods it is worth to acknowledge the strip methods proposed by [60] and by [10], or that developed by SISTEMAR [52] for the specific case of tip loaded propellers belonging to the Contracted and Tip Loaded (CLT) family. Some attempts to unify the scaling approaches have been proposed. Helma [28], for instance, proposed a scaling approach based on the concept of equivalent propeller entirely independent of the propeller geometry or blade loading, applicable, on the basis of the first proposed exemplary results, to any kind of propellers not experiencing flow separation.

Although the ITTC’78 Performance Prediction Method as well as other empirical scaling approaches were able to provide acceptable results in engineering applications, the increasing availability of unconventional propulsive configurations (i.e. CLT and Kappel propellers or Ducted propulsion systems) raises doubts about the reliability for instance of a statistical approach based on outdated geometries and profile shapes. The ITTC Conference, for its part, suggested that standard procedures for the extrapolation of model tests results cannot be applied to unconventional propellers [32] and recommended the development, by using dedicated experimental and numerical investigations, of updated scaling procedures suitable also for unconventional geometries. The first attempt following this advice was the ITTC Benchmarking Test Case [33] which was proposed to investigate the role of scale effects on a conventional (the PPTC Propeller VP1304, [6,24,26]) and an unconventional (the Tip Rake Propeller P1727 [25]) propellers using numerical calculations. The rapid development of Computational Fluid Dynamics methods, such Reynolds Navier-Stokes (RANS) equations, indeed, provided reliable tools for propeller performance predictions. The recent developments in turbulence modelling allow for the inclusion of laminar to turbulent transition phenomena in the calculations: the possibility to solve accurately the flow at different Reynolds number regimes, definitely, offers an alternative, reliable and flexible scaling method.

Using CFD for estimating propellers performance, scaling rules and the influence of some geometrical characteristics on full scale functioning is a relatively recent activity. Abdel-Maksoud and Heinke [1] carried out detailed CFD investigation to study the scale effects on open water characteristics of ducted propellers while Funeno [15] analysed the scale effect and its influence on hub vortex of open propellers. Similar analyses were carried out by Stanier [58], Bulten and Nijland [11], Krasilnikov et al. [34] with the extension to the study of the influence of the blade skew distribution using ANSYS Fluent and the SST $k - ω$ turbulence model. The role of high Reynolds number on propeller performance was investigated in [55] in the case of a tip loaded CLT propeller. The same geometry was then used to compare RANS results, usual ITTC’78 performance predictions, semi-empirical corrections, strip and BEM based methods with full scale sea trials [22]. Muller et al. [49] proposed a scaling approach based on numerical calculations using ANSYS CFX. Rijpkema et al. [53] analysed model scale propellers using two different fully turbulent two-equation models. They showed that the limiting streamlines from RANS were more circumferentially oriented with respect to paint tests, which indicates an overestimation of the size of the turbulent boundary layer, and observed a substantial difference in predicted performance compared with model tests carried out with smooth (without the application of leading edge roughness) geometries. Accurate model scale predictions (then, reliable scaling approaches based on numerical analyses) can not be achieved, as in the aforementioned literature, using turbulence models relevant only for fully developed turbulent flows since most of the model scale tests clearly lies in the critical Reynolds range and laminar flow could exist over a large portion (70%) of the blade span [15]. Then, the use of turbulence models able to deal with laminar to turbulent transition is mandatory.

Among transitional sensitive turbulence models, the most promising for applications to marine propellers are those based on the “single-point” concept, i.e. those which, by not requiring integral or non-local information (boundary layer momentum thickness or downstream distance from the leading edge) to derive the transition point, facilitate their application in the case of the unstructured meshes necessary for complex geometry discretizations. The $γ - {Re}_{θ}$ correlation based model [36–38,46] and the $k - k_{L} - ω$ model [64,67] based on the concept of laminar kinetic energy [42] are probably the most popular approaches.

The $γ - {Re}_{θ}$ , in particular, was broadly applied since the publication by Malan et al. [40] of the key model correlations, kept unpublished by the original authors, and of a tuning approach necessary to calibrate the model constants depending on the possible small implementation differences that might exist between different CFD codes, such as variants of the base turbulence model. One of the first application of this model to marine propellers is that by Muller et al. [49], closely followed by the application of the same model in the case of open [5,69], ducted [7,9] and tip loaded [48] propellers. Results showed a substantial improvement in the prediction of the model scale propeller performance, with good correlations also in terms of predicted limiting streamlines on propeller blade (i.e. prediction of the transition occurrence and location) if compared to available paint tests.

The application of the $k - k_{L} - ω$ model to propeller was not so widespread and only few tests, mainly from the developers of the model, were recognized in literature. For instance, Wang and Walters [68] proposed calculations with this transition sensitive model for the P5168 propeller. Their calculations, however, shown laminar to turbulent transition on the propeller blades in correspondence of a chord-based Reynolds number of about $5 \cdot 10^{6}$ , which is significantly higher than the usual critical Reynolds number range for laminar to turbulent transition in case of model scale propellers. Helal et al. [27], similarly, computed the performance of the well-known INSEAN E779A propeller using the implementation of this model available in ANSYS Fluent. The $k - k_{L} - ω$ model is available in the OpenFOAM package since version 2.1 [61] and its formulation was subjected to substantial modifications to comply with the corrections proposed by Furst [16] to overcome the inconsistencies erroneously published in the reference paper [64]. With the updated formulation, the model has shown a very good agreement with simple flat plate tests, both in terms of velocity distributions across the boundary layer and distribution of the frictional coefficient. Unfortunately, also a certain sensitivity to inlet quantities and a tendency to postpone the transition in case of flows with adverse pressure gradients was observed.

The ability to predict, at first even only qualitatively, the role of transition on propeller performance (usually computed in fully turbulent flows), can have a significant impact in modern propeller analyses and designs. The aim of the present work, in the light of this, is to verify the possible improvements in propeller performance predictions in model scale using the RANS equations with the $k - k_{L} - ω$ transition sensitive turbulence model of the OpenFOAM library, version 4.1 [62]. Results are presented for three different test case, for which model scale measurements in open water are available: the first two are the propellers selected for the ITTC Benchmarking Test Case [24,25] while the third is a propeller from the AQUO collaborative research project supported by the European Commission [3]. Mesh sensitivity, as well as the influence of inlet turbulent quantities are investigated, considering at first the simple flat plate test case T3A from the ERCOFTAC database [13] for a preliminary validation of the current implementation of the transition sensitive turbulence model. In the absence of dedicated paint tests, the outcomes of present calculations with the transition sensitive turbulence model are comparatively discussed with fully turbulent calculations using the SST $k - ω$ turbulence model, the data collected during the 28th ITTC Conference [33] and the additional calculations available [48,69] for the same test cases. Results are discussed in order to evaluate the ability of the model to capture the importance of boundary layer transition in the prediction of propeller flows and to evidence for the need to account for transition phenomena in model scale calculations. By the comparison with similar calculations available in literature, the aim is also to highlight the differences with respect to alternative formulations (the $γ - {Re}_{θ}$ ) and the possible limitations of the proposed turbulence model, which was already shown to be less successful in predicting some complex flow [66] and to be excessively sensible to freestream turbulence.

2. Test cases: VP1304, P1727 and P2772 propellers

In present study, three propeller have been considered. The first, the VP1304, is a conventional propeller already published by the SVA Potsdam in the course of the Propeller Workshop under the acronym PPTC (Potsdam Propeller Test Case) held at the 2nd Symposium on Marine Propulsors in Hamburg [6,26]. It is a five-bladed, right-handed, controllable-pitch propeller, designed by SVA in 1998. The second geometry, the P1727, is an unconventional propeller designed itself by SVA Potsdam for the german funded research project “Tip Rake”. It is a four-bladed, right-handed, fixed-pitch propeller, characterized by a finite chord at tip. With respect to a conventional geometry, the outer radial part of the blade is “curved” (by a specific distribution of the rake) towards the pressure side. The resulting geometry resembles, for what regards both shape and functioning principles, that of traditional CLT [17,51] and of new generation of CLT [18,23]. Both were selected by the 28th ITTC Propulsion Committee for a benchmark [32,33] devoted to the assessment of updated scaling methods for propeller performance in case of unconventional geometries like Kappel, Tip Loaded and CLT. Model scale measurements are available by SVA as a part of the ITTC Benchmarking Test Case. The third one, the P2772 is the four-bladed, controllable-pitch, left-handed propeller of the Medium Size Tanker provided in the framework of the European Project AQUO [3]. Model scale tests are available from SSPA Sweden [3].

Fig. 1.

Propeller test cases.

Table 1

Main characteristics and functioning conditions of the test case propellers

Main characteristic			VP1304	P1727	P2772
Model Diameter	D	[mm]	250	238.6	240
Pitch at $r / R = 0.7$	$P / D_{0.7}$	[-]	1.635	0.841	0.870
Chord at $r / R = 0.7$	$c / D_{0.7}$	[-]	0.417	0.236	0.291
Skew at tip	$θ_{tip}$	[°]	18.8	25.7	26
Area ratio	$A_{E} / A_{O}$	[-]	0.779	0.444	0.450
Hub Diameter Ratio	$d_{h} / D$	[-]	0.3	0.154	0.279
Number of blades	Z	[-]	5	4	4
Propeller rate of revolution	$rps$	[1/s]	15	18	15
Advance coefficient	J	[-]	0.6–1.4	0.1–0.9	0.1–0.9
Reynolds number $(\cdot 10^{5})$ at $r / R = 0.7$	${Re}_{0.7}$	[-]	7.8–8.9	4.6–5.0	4.8–5.3

An overview of the propellers geometry is given in Fig. 1; main geometrical data and functioning conditions considered for present calculations are summarized in Table 1. Chord-based Reynolds numbers (reference at $r / R = 0.7$ ) during model scale tests are within the critical range (in particular for P1727 and P2772), then a significant influence of laminar to turbulent transition phenomena can be expected, making these test cases suitable for the validation of the transition modelling using numerical approaches.

3. Models

3.1. RANS equations

Under the hypotheses of incompressible and steady flow, the time-averaged continuity and momentum equations (RANS) can be written in the differential form proposed in Equation (1), assuming $x_{i = 1, 2, 3}$ as the Cartesian coordinates and $V_{i = 1, 2, 3}$ the Cartesian components of the velocity vector in the body-fixed reference frame: $\begin{matrix} (1) & \{\begin{matrix} \frac{\partial V_{i}}{\partial x_{i}} = 0 \\ ρ V_{j} \frac{\partial V_{i}}{\partial x_{j}} = - \frac{\partial p}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} [μ (\frac{\partial V_{i}}{\partial x_{j}} + \frac{\partial V_{j}}{\partial x_{i}})] + \frac{\partial τ_{i j}}{\partial x_{j}} \end{matrix} \end{matrix}$

The fluid static pressure and density are, respectively, p and ρ, μ is the fluid viscosity and $τ_{i j}$ are the Reynolds stresses resulting from the averaging process of the momentum equations, which are treated on the basis of the Boussinesq hypothesis, then proportional to the trace-less mean strain rate tensor times the eddy viscosity $μ_{T}$ . Modelling the eddy viscosity using simplified relationship is the key point of any turbulence model. The SST $k - ω$ [44] and the $k - k_{L} - ω$ [64] models used in present analyses are briefly illustrated in Sections 3.2 and 3.3. A Moving Reference Frame, with the inclusion of the relative source terms in Equations (1), finally allowed for steady-state calculations.

3.2. The SST $k - ω$ turbulence model

Among the available turbulence models for fully turbulent flow, the SST $k - ω$ two-equations model of Menter [44,45] was used. The Menter and Esch [43] formulation with the updated coefficients and the corrections for the production terms $α ρ / μ_{t} \tilde{P_{k}}$ based on [45,50] is the implementation currently adopted in the OpenFOAM library. This model makes use of two equations, one for the turbulence kinetic energy k and one for the specific dissipation rate ω. In the inner part of the boundary layer the formulation is that of a standard $k - ω$ model, which makes the $SST$ model usable all the way down to the wall through the viscous sub-layer without any extra damping functions; in the free stream the formulation switches to a $k - ϵ$ like behaviour, avoiding the excessive sensitiveness to the inlet free-stream turbulence quantities typical of the standard $k - ω$ approach. $\begin{matrix} (2) & \{\begin{matrix} ρ V_{j} \frac{\partial k}{\partial x_{j}} = \tilde{P_{k}} - β^{*} ρ ω k + \frac{\partial}{\partial x_{j}} [(μ + α_{k} μ_{t}) \frac{\partial k}{\partial x_{j}}] \\ ρ V_{j} \frac{\partial ω}{\partial x_{j}} = \frac{α ρ}{μ_{t}} \tilde{P_{k}} - β ρ ω^{2} + \frac{\partial}{\partial x_{j}} [(μ + α_{ω} μ_{t}) \frac{\partial ω}{\partial x_{j}}] \\ + 2 (1 - F_{1}) ρ α_{ω 2} \frac{1}{ω} \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}} \end{matrix} \end{matrix}$

The eddy viscosity is computed according to: $\begin{matrix} (3) & μ_{t} = a_{1} \frac{k}{max (a_{1} ω, b_{1} F_{23} S)} \end{matrix}$ where $S$ is the strain rate magnitude, $a_{1}$ and $b_{1}$ two model constants (0.31 and 1, respectively) and $F_{23}$ is a blending function in accordance with the modification to the original SST $k - ω$ model proposed in [43,45]. A detailed description of the current SST $k - ω$ model, of its model constants and of the blending functions used to couple the inner $k - ω$ with the outer $k - ϵ$ regions can be found in [62].

3.3. The $k - k_{L} - ω$ transition model

The $k - k_{L} - ω$ is a turbulence model developed by Walters et al. [64,65,67] to reproduce the transition process. It is a single-point model in the sense that it does not require integral or non-local information to be included in the model for the prediction of the transition location and it is a “phenomenological” model since it attempts to model the physics of transition rather than recurring to a purely empirical-based correlations. Since the physics of transition is still not entirely understood [46], correlation-based models sometimes have been preferred, but recent advancements underlying the physical mechanisms as well as the universal characteristics of boundary layer flows allowed for reasonably accurate models, based on a better knowledge of the relevant scaling mechanisms. This model is the development of the formulation proposed in [65,67], which originally made use of a $k - ϵ$ formulation for the Reynolds stresses. The $k - k_{L} - ω$ is a three-equation model (Equation (4)) with transport equations solved for the turbulent kinetic energy (k), the specific dissipation rate (ω) and the laminar kinetic energy ( $k_{L}$ ) which is a concept first introduced by Mayle and Schulz [42]. The laminar kinetic energy is used to represent the energy of velocity fluctuation modes observable in the boundary layer prior to transition, which amplifications lead to transition and to turbulent flow. These fluctuations are related to Tollmien–Schlichting waves, which are precursor to natural transition and to Klebanoff modes which characterize the bypass transition. Transition (either natural or bypass, as from the last formulation of the model proposed in [64,68]) is represented as a transfer of energy from laminar to turbulent kinetic energy. Laminar kinetic energy is produced by the interaction of the Reynolds stresses associated with non-turbulent velocity fluctuations and mean shear while turbulent kinetic energy represents the magnitude of fluctuations associated to turbulent flows due to strong three dimensionality, viscous dissipation and energy cascade. In the light of LES simulations of the transitional boundary layer [39] this seems to be a more physically correct production process with respect to the transport of energy from the freestream into the boundary layer due to the pressure diffusion term in the kinetic energy budget proposed by [42].

The various terms of the equations represent production, destruction, transport and diffusion and are described in details, as well as the constants of the model, in [64]. The corrections to the wall-limited turbulence length scale, to the intermittency factor, to the dissipation and diffusion terms and to some of the coefficient proposed by Furst [16] are used in the current implementation of the model in the OpenFOAM library [62]. $\begin{matrix} (4) & \{\begin{matrix} ρ V_{j} \frac{\partial k}{\partial x_{j}} = ρ (P_{k} + R_{B P} + R_{NAT} - ω k - D_{T}) + \frac{\partial}{\partial x_{j}} [(μ + \frac{ρ α_{T}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}] \\ ρ V_{j} \frac{\partial k_{L}}{\partial x_{j}} = ρ (P_{k L} - R_{B P} - R_{NAT} - D_{L}) + \frac{\partial}{\partial x_{j}} [μ \frac{\partial k_{L}}{\partial x_{j}}] \\ ρ V_{j} \frac{\partial ω}{\partial x_{j}} = ρ [C_{ω 1} \frac{ω}{k} P_{k} + (\frac{C_{ω R}}{f_{W}} - 1) \frac{ω}{k} (R_{B P} + R_{NAT}) \\ - C_{ω 2} ω^{2} + C_{ω 3} f_{ω} α_{T} f_{W}^{2} \frac{\sqrt{k}}{d^{3}}] + \frac{\partial}{\partial x_{j}} [(μ + \frac{ρ α_{T}}{σ_{ω}}) \frac{\partial ω}{\partial x_{j}}] \end{matrix} \end{matrix}$

4. The ERCOFTAC T3A test case

The OpenFOAM implementation of the $k - k_{L} - ω$ turbulence model for transition prediction has been first tested for a simple zero pressure gradient flat plate in order to verify the response to the freestream turbulence (i.e. inlet turbulence quantities) and to compare with measured data the predicted location of the transition point. The test case belongs to the T3 validation cases family from the European Research Consortium on Flow, Turbulence and Combustion (ERCOFTAC), which are recognized a standard for this kind of analyses. In particular, the T3A case (turbulent intensity at flat plate leading edge equal to 3.3%) was considered for the validation [13]. The test case is the same used in [64] and [16] to check the implementation of the transitional model in ANSYS Fluent 6.3 and in OpenFOAM respectively. Here it is proposed again to verify its accuracy and to discuss the sensitivity to inlet parameters.

The computational domain was slightly simplified with respect to that of experiments. For instance, the rounded leading edge of the flat plate was not considered. Boundary conditions, summarized in Fig. 2, were selected to match as closely as possible the experimental geometry and to comply with the assumption of the model. For instance, the wall boundary condition for the specific dissipation rate ω is substantially different from the one commonly used in the $k - ω$ class of turbulence models since, as proposed for low-Reynolds number $k - ϵ$ models, the increase of the viscous dissipation at walls is directly incorporated into the equations for the kinetic energies (laminar and turbulent) [64]. The same boundary conditions, obviously, has been used at walls in case of propeller analyses (Table 4). A symmetry condition was applied on the bottom surface upstream of the flat plate leading edge as well as on the top surface of the computational domain. Left and right patches were set respectively as velocity inlet and pressure outlet. The dimensions of the computational domain were set equal to those used by Furst [16] in order to ensure negligible acceleration of the freestream due to the development of the boundary layer. The reference domain, then, consists in a rectangular region of $[- 0.5, 2.9] \times [0, 0.175] m$ (the flat plate starting at $x = 0 m$ ), discretized with a body-fitted structured mesh realized with 700 cells along the longitudinal coordinate (200 in the inlet region) and 105 in the vertical directions. Refinements, by appropriate grading of the mesh spacing, were placed close to the inlet edge of the flat plate and in the wall normal direction for an average non-dimensional wall distance $y^{+}$ of about 0.2. From this reference configuration, additional computational domains were derived in order to assess the sensitivity to the inlet (position and turbulence quantities) and the tolerance to non-dimensional wall distances greater than 1.

Fig. 2.

Mesh arrangement and boundary condition for the zero pressure gradient flat plate. Reference geometry ( $x_{inlet} = - 0.5 m$ ).

The sensitivity to inlet was verified with three additional position of the inlet (respectively $x_{inlet} = - 0.25, - 1.0$ and −2.0 m). Correspondent meshes were defined by successive halving or doubling of the reference spacing of the inlet region, in order to ensure the same spacing and grading between adjacent cells.

The influence of inlet quantities on transition (and their correlation with the inlet position) was checked by computing the boundary layers quantities for any combination of inlet turbulence intensity ( $I_{% inlet} = [1, 2, 3, 5, 10]$ ) and turbulent viscosity ratio ( $μ_{T} / μ = [1, 10, 50, 100, 500, 1000, 2000]$ ) for each geometrical configurations. Since it is important to pay the proper attention to the turbulent specification on inflow boundaries to encounter realistic turbulence intensities in the boundary layers within the transition zone, care must be taken in the choice of the necessary combination of turbulent viscosity ratio (or turbulent length scale) and turbulent intensity. The rate of decay of the turbulent intensity indeed, is relatively high [57]. If, in principle, higher values of turbulent viscosity at inlet could be preferred in order to mitigate the turbulence intensity decay and provide, from a potentially very far inlet, the right amount of turbulence in the transition region, also the significantly smaller rate of decay of the effective turbulent viscosity should be considered. Inlet values of (high) eddy viscosity can easily propagate far downstream, with the obvious consequence of excessive diffusion, which has to be taken into account for reliable prediction of flow fields and performance.

Finally, the tolerance with respect to non-dimensional wall distances higher than 1 was verified, instead, only for the reference configuration by changing the grading of the cells in the normal direction. A total of ten cases, from $y^{+} ≃ 0.2$ to $y^{+} ≃ 12$ were considered.

Fig. 3.

Influence of turbulent quantities at inlet, friction coefficient distributions and decay rates varying inlet position with the calibrated inlet parameters ( $I_{% inlet}$ and $μ_{T} / μ_{inlet}$ ) of Table 2 for the ERCOFTAC T3A test case. Transition sensitive calculations with $y^{+} = 0.2$ .

The influence of the turbulence inlet quantities on transition location is summarized in Fig. 3. The flow characteristics on the flat plate for any given inlet location were calculated for all of the 35 possible combinations of turbulence intensity and turbulence viscosity ratio at inlet. An example ( $I_{% inlet} = 10 %$ , $x_{inlet} = - 0.25 m$ ) is shown in Fig. 3(a). Then, by analyzing the kinetic energy decay alongside the computational domain, an additional set of calculations (Fig. 3(c)) was carried out by considering only the combinations (listed in Table 2 and shown in Fig. 3(b)) of these quantities necessary to achieve, at the flat plate leading edge, exactly the turbulence intensity measured during experiments (3.3%). The couple of turbulent intensity and viscosity ratio at inlet identified for the reference test case (case 2B, $x_{inlet} = - 0.5 m$ ) was finally used for the analysis of the influence of the first boundary layer cell thickness, which is summarized in Fig. 6.

Table 2

Turbulence intensity and turbulence viscosity ratio combinations at inlet for a given turbulent intensity (3.3%) at the flat plate leading edge

Case	$x_{inlet}$ [m]	$I_{% inlet}$	$μ_{T} / μ_{inlet}$	$I_{% @ flat plate L E}$
case 1A	−0.25	5%	290	3.37%
case 1B	−0.25	10%	190	3.35%
case 2A	−0.50	5%	535	3.29%
case 2B	−0.50	10%	380	3.33%
case 3A	−1.00	5%	1080	3.30%
case 3B	−1.00	10%	735	3.29%
case 4A	−2.00	5%	2140	3.28%
case 4B	−2.00	10%	1500	3.31%

Fig. 4.

Role of the turbulence intensity decay and of its value at the flat plate leading edge on the friction coefficient distributions for the ERCOFTAC T3A test case. Transition sensitive calculations with $y^{+} = 0.2$ .

Computed results, which currently are limited to the skin friction coefficient distributions along the flat plate, are in good agreement with the ERCOFTAC database. As expected from the analytical solution of the convective equations for k and ω of the SST $k - ω$ turbulence model proposed in [57], both the increase of the inlet turbulence and of the turbulent viscosity ratio leads towards an early laminar to turbulent transition. The increase of both these parameters is effective in reducing the decay of the turbulent kinetic energy and allowing for higher value of turbulent intensity which obviously stimulates the transition to turbulent flow. Once these parameters are selected in order to have almost the same turbulent intensity at the flat plate leading edge (Fig. 3(b)), the results are almost overlapped regardless the inlet location and the local decay behaviour of the turbulent intensity. There are only two configurations ( $x_{inlet} = - 0.25$ , $I_{inlet %} = 10 %$ and $x_{inlet} = - 2.00, I_{% inlet} = 10 %$ ) which show a certain discrepancy with respect to measurements but a clear correlation with the decay of the turbulence intensity from flat plate leading edge to the zone where transition is expected cannot be recognized. Both these two configurations show an earlier transition which, however, in one case ( $x_{inlet} = - 0.25, I_{inlet %} = 10 %$ ) corresponds to a sensibly lower value of local turbulence intensity in the transition region, in the other ( $x_{inlet} = - 2.00$ , $I_{% inlet} = 10 %$ ) to a value comparable with those of cases showing a better prediction of the transition point.

For reliable predictions of the transition point, the prevailing importance of the turbulence intensity level at the leading edge, at least in the specific case of the flat plate, is underlined in Fig. 4. In this case, among the calculations used for the calibration of the turbulence intensity at the flat plate leading edge, three configurations have been selected. They have, locally at the leading edge, very similar decay rates and, across the zone where transition experimentally takes place, they exhibit values of turbulent intensity between the minimum (1.5%) and the maximum (3%) observed in the case of the calculations with the calibrated parameters of Fig. 3(b). The corresponding turbulence intensity at the flat plate leading edge is close to the experimental value of 3.3% only in one case ( $x_{inlet} = - 2.00$ , $I_{% inlet} = 5 %$ , $μ_{T} / μ_{inlet} = 2000$ ), which is exactly the only with a reasonable transition prediction. The behaviour, in terms of predicted skin friction coefficient distributions of the other two cases, is very different from what observed in Fig. 3(c), where the turbulence intensity was prescribed at the flat plate leading edge regardless the decay ratio from the leading edge to the transition point and, consequently, from its resulting value across the transition zone. Even if in the transition zone the turbulence intensities are comparable to those of Fig. 3(b) (in that case achieved starting with the right intensity at the leading edge), transition is sensibly postponed: for this particular and geometrically simple case, it seems then that the leading edge value of turbulent intensity (rather than the local decay) is the main responsible of the predicted transition location.

The behaviour of the laminar to turbulent transition phenomenon, also in the light of the application of this model for overall propeller performance prediction, is more than satisfactory. At higher Reynolds numbers (Fig. 5), which resemble full scale/fully turbulent analyses, the friction coefficient distributions by the $k - k_{L} - ω$ turbulence model smoothly converge, regardless the transition point, to the fully turbulent friction distribution (1/7 power law and similar approximations valid for the considered range of local Reynolds number) proposed in [56] for the flat plate case. Fully turbulent calculations using the SST $k - ω$ model with ( $y^{+} < 1$ ) or without ( $y^{+} = 20$ ) wall functions converge, as expected, to the same values.

Fig. 5.

Friction coefficients by the $k - k_{L} - ω$ and the SST $k - ω$ turbulence models (with and without wall functions, respectively $y^{+} = 20$ and $y^{+} = 0.2$ ) at full-scale Reynolds numbers. Comparison with 1/7 power law and Schlichting empirical correlation [56].

The model, at least for this simple test case, seems reasonably robust and capable of dealing with meshes having average $y^{+}$ up to 3, as shown in Fig. 6. Calculations have been carried out for average non-dimensional wall distances of 0.2, 0.4, 0.6, 0.8, 1, 1.5, 2, 3, 8 and 12. For the higher values it is possible to observe a degradation of the prediction of the frictional coefficient especially in correspondence of the “laminar portion” at the leading edge of the flat plate and a certain overestimation with respect to usual empirical correlations [56] or calculations with the SST $k - ω$ model in correspondence to the fully developed turbulent boundary layer, even if the transition location is very similarly predicted. Instead, all the calculations with $y^{+} < 3$ are almost perfectly overlapped. A certain tolerance (even if verified only for a simple geometry) to values higher than the canonical $y^{+} = 1$ could result particularly important for the application to propellers, where the arrangement of a prism layer with a sufficiently high number of very thin cells at wall is not obvious, and each time the focus is on overall performance predictions rather than on local characteristics.

Fig. 6.

Influence of the non-dimensional wall distance (note that curves with $0.2 < y^{+} < 3$ are almost perfectly overlapped) on the friction coefficient distribution.

5. Propeller open-water results

For the three propeller test cases, calculations have been carried out to replicate the open water towing tank tests summarized in Table 1. By exploiting the periodicity of the geometry, only one blade passage was modelled using a steady-state simulation based on the simpleFOAM library in a relative rotating frame at constant rotational speed and appropriate boundary conditions to match the interfaces of the computational domain. The “consistent” version of the SIMPLE algorithm [62] was used to iteratively couple velocity and pressure fields. Under-relaxation factors were equal to 0.9 for the velocity and 0.6 for turbulent quantities. Convective terms were discretized with a second order, upwind-biased scheme (for the advection of velocity) and blended first-/second-order scheme (for turbulent quantities) while gradients were approximated with the Gauss quadrature and the multidimensional limiter [62]. Numerical schemes for Laplacian and surface normal gradients were selected in order to account for non-orthogonal meshes. Since measurements were carried out at the towing tank, the computational domain for numerical analyses was set-up to resemble the experimental facility and in particular to avoid any interference of the far field boundaries on the predicted propeller performance. For all the cases, as prescribed by the ITTC Benchmarking Test Case [33], the computational domain has a cross area at the propeller plane one hundred time higher than the area of the propeller disk ( $D_{domain} / D_{prop} = 10$ ). The inlet was placed four propeller diameters in front of the propeller disk, the outlet six propeller diameters aft and uniform inflow was assumed as the initial condition. A representation of the computational domain and of the boundary conditions are shown, respectively, in Fig. 7 and Tables 3, 4.

Fig. 7.

Computational domain for steady open water propeller performance predictions.

Table 3

Boundary conditions for OpenFOAM calculations. Fully turbulent analyses with the SST $k - ω$ turbulence model

Patch	Velocity [ms⁻¹]	Pressure [Pa]	k [m²s⁻²]	ω [s⁻¹]	$μ_{T}$ [Pa · s]
Inlet	Based on J	$\frac{\partial p}{\partial n} = 0$	$\propto I_{%}$	$\propto I_{%}$	$\propto k$ and ω
Outlet	$\frac{\partial U}{\partial n} = 0$	$p_{ref}$	$\frac{\partial k}{\partial n} = 0$	$\frac{\partial ω}{\partial n} = 0$	$\propto k$ and ω
Outer Boundary	$\frac{\partial U}{\partial n} = 0$	$p_{ref}$	$\frac{\partial k}{\partial n} = 0$	$\frac{\partial ω}{\partial n} = 0$	$\propto k$ and ω
Blade	No-slip	$\frac{\partial p}{\partial n} = 0$	Wall func.	Wall func.	Wall func.
Shaft	No-slip	$\frac{\partial p}{\partial n} = 0$	Wall func.	Wall func.	Wall func.

Table 4

Boundary conditions for OpenFOAM calculations. Transition enabled analyses with the $k - k_{L} - ω$ turbulence model

Patch	Velocity [ms⁻¹]	Pressure [Pa]	k [m²s⁻²]	$k_{L}$ [m²s⁻²]	ω [s⁻¹]	$μ_{T}$ [Pa · s]
Inlet	Based on J	$\frac{\partial p}{\partial n} = 0$	$\propto I_{%}$	0	$\propto I_{%}$	$\propto k$ and ω
Outlet	$\frac{\partial U}{\partial n} = 0$	$p_{ref}$	$\frac{\partial k}{\partial n} = 0$	$\frac{\partial k_{L}}{\partial n} = 0$	$\frac{\partial ω}{\partial n} = 0$	$\propto k$ and ω
Outer Boundary	$\frac{\partial U}{\partial n} = 0$	$p_{ref}$	$\frac{\partial k}{\partial n} = 0$	$\frac{\partial k_{L}}{\partial n} = 0$	$\frac{\partial ω}{\partial n} = 0$	$\propto k$ and ω
Blade	No-slip	$\frac{\partial p}{\partial n} = 0$	0	0	$\frac{\partial ω}{\partial n} = 0$	0
Shaft	No-slip	$\frac{\partial p}{\partial n} = 0$	0	0	$\frac{\partial ω}{\partial n} = 0$	0

Results have been collected after appropriate mesh sensitivity analyses in order to limit, as much as possible, the influence of the spatial discretization and attempt to discuss the net influence of turbulence modelling and transition on predicted performance. In transitional flow, indeed, the location of the transition or of the laminar separation bubbles (and then their influence on predicted performance) is expected to be more sensitive to grid resolution with respect to fully turbulent conditions. To this aim, four meshes have been proposed for each test case (Very Coarse, Coarse, Medium, Fine), obtained by reducing or increasing by a factor of 1.4 the size (side length) of the reference cell of the Medium mesh configuration, shown, for P2772, in Fig. 8. Smaller edge spacing were applied to blade leading, trailing and tip regions while the hierarchical nature of the local and zonal refinements allowed for similar meshes, with the only caution to maintain, for any mesh choice, the non-dimensional wall distance below the critical value of the turbulence model to ensure a proper viscous sublayer resolution. Prism cells near the wall surfaces were generated accordingly with a normal spacing increasing geometrically with a growth rate of 1.3. A summary of the grid parameters is reported in Table 5. Both fully turbulent and transition sensitive calculations were carried out with a resulting average $y^{+}$ lower than 0.5 (at the design advance coefficient) and only a limited portion of the blade (mainly at suction side leading edge, where velocities are higher) with a non-dimensional wall distance higher than 1 ( $y^{+} ≃ 2$ ). In the light of the calculations carried out for the flat plate, this arrangement and the resulting dimensionless distance were considered acceptable and indispensable for reasonable computational efforts. Fully turbulent analyses with the SST $k - ω$ turbulence model were carried out using the family of blended wall functions (kqRWallFunction, omegaWallFunction and nutUSpaldingWallFunction, for k, ω and $μ_{T}$ respectively) available in OpenFOAM [62].

Fig. 8.

Surface and volume mesh details. Reference mesh of the P2772 propeller.

Table 5

Surface and volume mesh parameters

Mesh type	Hexa-dominant
Avg. Surface mesh size (Blade)	0.4% of propeller Diameter
Avg. Surface mesh size (Shaft)	0.4% of propeller Diameter
Minimum Surface mesh size (LE, TE, Tip)	0.04% of propeller Diameter
Avg. cell side length (Propeller zone)	1% of propeller Diameter
Avg. cell side length (Wake zone)	2% of propeller Diameter
Number of prism layers	17
Prism layer thickness (Blade)	1.2 mm
Avg. $y^{+}$ on Blade	0.5

5.1. Influence of turbulence parameters on propeller performance and trailing wake: fully turbulent analysis

The analysis of the flat plate test case demonstrated that the transition point can be successfully predicted by providing the right amount of turbulence in the transition region but no information circa the influence of relatively high values of eddy viscosity on other flow characteristics (the propeller wake, for instance) could be derived from those analyses. For this aim, the VP1304 test case was used for a preliminary investigation of the influence of the turbulence quantities at inlet on propeller performance. At the design advance coefficient of 1.2, thrust and torque coefficients were calculated for 20 combinations of turbulence intensity at inlet ( $I_{% inlet} = [1, 2, 3, 5, 10]$ ) and turbulence viscosity ratio ( $μ_{T} / μ = [10, 50, 100, 500, 1000]$ ). These analyses were carried out in fully turbulent conditions (since the influence of turbulent quantities at inlet on propeller trailing wake can be verified also in fully turbulent flow and dedicated analyses to verify the influence of turbulence on transition are proposed for each propeller) and with the Medium mesh arrangement since the investigation is useful also only in a comparative sense.

Fig. 9.

Comparison of VP1304 propeller performance with respect to turbulence quantities at inlet. SST $k - ω$ calculations at the design advance coefficient $J = 1.2$ .

Results are collected in the diagrams of Fig. 9 while a detailed comparison of the propeller trailing wakes at different turbulence conditions is proposed in Figs 10, 11 and 12. In terms of propeller performance, the influence of turbulence quantities is almost negligible. A certain influence is appreciable only for combinations of high values of turbulence intensity and turbulent viscosity ratio. For both thrust and torque, indeed, the variations of propeller performance with respect to those computed with usual choices ( $I_{% inlet} = 1$ , $μ_{T} / μ = 10$ ) are negligible (less than 0.1%) for any value of the turbulent viscosity ratio if the turbulence intensity is kept below 5%. Only for turbulence intensities higher than 5% and turbulent viscosity ratio greater than 500, propeller performance show a certain variation, which however is always below 1%.

Fig. 10.

Comparison of propeller wakes at $x / D = 0.2$ . Non-dimensional axial velocity contours.

Fig. 11.

Comparison of propeller wakes at $x / D = 1.0$ . Non-dimensional axial velocity contours.

Fig. 12.

Comparison of propeller wakes at $x / D = 1.0$ , $r / R = 0.7$ for different turbulence viscosity ratios.

Different conclusions arise when the influence of turbulence quantities is monitored through the analysis of the propeller trailing wake. Close to the propeller ( $x / D = 0.2$ as in Fig. 10) the diffusion induced by turbulence is quite low: calculations clearly show the trace of the blade trailing wake and the tip vortex, without any significant difference among the various choice of turbulence quantities. Moving the attention to a section far from the propeller disk ( $x / D = 1.0$ , Fig. 11), instead, the role of turbulence dissipation is clear. In addition to the smearing induced by the spatial discretization, which was not specifically prepared to deal with the accurate predictions of the flow field far from the propeller, it is possible to recognize the significant role of the turbulence viscosity. The higher is the turbulent viscosity ratio at inlet, the slower is the decay of the effective turbulence in the computational domain, which produces significantly smeared velocity distributions, as shown in Fig. 12. Until the turbulent viscosity ratio is small (<100), axial velocity distributions, as well as the position of peaks and troughs, are similarly predicted for any choice of the turbulent intensity. Higher values, on the contrary, induce a substantial dissipation which, depending on the applications (e.g. propeller/rudder interaction and induced pressure pulses) cannot be accepted for accurate predictions.

5.2. Propeller performance under transition conditions: Sensitivity to inlet quantities

Prior to the discussion of the influence of transition modelling (and the influence of inlet quantities) on propeller performance, an estimation of the discretization error (that related to the discrete representation of the governing equations) has been carried out through a numerical uncertainty analyses based on the Richardson extrapolation [12]. Results have been collected for the four different meshes described above, approximatively ranging from 1.6 to 11 Million cells for all the three test cases. Extrapolations and sensitivity indexes are shown in Tables 6, 7 and 8, for both fully turbulent and transition sensitive analyses, using the Coarse, the Medium and the Fine mesh arrangements only since in most of the cases the Very Coarse grid was too coarse and not in the asymptotic range (ratio between successive Grid Convergence Indexes [54] far form 1).

Table 6
Mesh sensitivity analysis for the VP1304 test case. Fully turbulent and transition sensitive calculations ( $I_{% inlet} = 3 %$ , $μ_{T} / μ_{inlet} = 600$ ) at $J = 1.2$

Turbulent Transition

$K_{T}$ $10 K_{Q}$ $K_{T}$ $10 K_{Q}$

Very Coarse 0.2662 0.7516 0.2916 0.7818

Coarse 0.2618 0.7383 0.2891 0.7751

Medium 0.2588 0.7263 0.2881 0.7717

Fine 0.2581 0.7219 0.2883 0.7721

Extrapolated value 0.2579 0.7193 0.2884 0.7722

Approximated relative error, $e_{a}^{medium-fine}$ 0.25% 0.61% 0.08% 0.06%

Extrapolated relative error, $e_{ext}^{medium-fine}$ 0.07% 0.36% 0.03% 0.01%

Fine grid convergence index, $G C I_{fine}^{medium-fine}$ 0.08% 0.45% 0.03% 0.01%

Apparent order of convergence, p 4.61 2.95 4.28 6.06

	Turbulent	Transition
Very Coarse	0.2662	0.7516	0.2916	0.7818
Coarse	0.2618	0.7383	0.2891	0.7751
Medium	0.2588	0.7263	0.2881	0.7717
Fine	0.2581	0.7219	0.2883	0.7721
Extrapolated value	0.2579	0.7193	0.2884	0.7722
Approximated relative error, $e_{a}^{medium-fine}$	0.25%	0.61%	0.08%	0.06%
Extrapolated relative error, $e_{ext}^{medium-fine}$	0.07%	0.36%	0.03%	0.01%
Fine grid convergence index, $G C I_{fine}^{medium-fine}$	0.08%	0.45%	0.03%	0.01%
Apparent order of convergence, p	4.61	2.95	4.28	6.06

Table 7

Mesh sensitivity analysis for the P1727 test case. Fully turbulent and transition sensitive calculations ( $I_{% inlet} = 3 %$ , $μ_{T} / μ_{inlet} = 400$ ) at $J = 0.5$

	Turbulent		Transition

	$K_{T}$	$10 K_{Q}$	$K_{T}$	$10 K_{Q}$
Very Coarse	0.1941	0.2669	0.2061	0.2722
Coarse	0.1947	0.2672	0.2078	0.2743
Medium	0.1959	0.2685	0.2089	0.2752
Fine	0.1959	0.2684	0.2080	0.2737
Extrapolated value	0.1959	0.2683	0.2031	0.2711
Approximated relative error, $e_{a}^{medium-fine}$	0.02%	0.06%	0.42%	0.55%
Extrapolated relative error, $e_{ext}^{medium-fine}$	0.00%	0.01%	2.40%	0.97%
Fine grid convergence index, $G C I_{fine}^{medium-fine}$	0.00%	0.01%	2.93%	1.19%
Apparent order of convergence, p	9.50	6.01	0.49	1.36

Table 8

Mesh sensitivity analysis for the P2772 test case. Fully turbulent and transition sensitive calculations ( $I_{% inlet} = 3 %$ , $μ_{T} / μ_{inlet} = 300$ ) at $J = 0.5$

	Turbulent		Transition

	$K_{T}$	$10 K_{Q}$	$K_{T}$	$10 K_{Q}$
Very Coarse	0.1989	0.2957	0.2174	0.3154
Coarse	0.1988	0.2954	0.2163	0.3140
Medium	0.1980	0.2939	0.2165	0.3155
Fine	0.1978	0.2936	0.2164	0.3160
Extrapolated value	0.1977	0.2935	0.2163	0.3162
Approximated relative error, $e_{a}^{medium-fine}$	0.10%	0.10%	0.05%	0.16%
Extrapolated relative error, $e_{ext}^{medium-fine}$	0.03%	0.03%	0.03%	0.08%
Fine grid convergence index, $G C I_{fine}^{medium-fine}$	0.04%	0.03%	0.04%	0.10%
Apparent order of convergence, p	4.12	4.70	2.69	3.32

In the case of fully turbulent calculations (using the SST $k - ω$ model), the outcomes of the sensitivity analyses agree well with the convergence rate observed in similar applications with the same solver [19]. With the exception of the torque coefficient in the case of the P1727 propeller, which shows a slightly oscillatory convergence, propellers performance have a good monotonic convergence trend demonstrated by the particularly low values of the Grid Convergence Index measuring the change in predictions due to additional mesh refinements. Also uncertainties, monitored through the approximated and the extrapolated relative error, are low (less than 1%). In the case of transition sensitive calculations, results from the sensitivity analyses are similar, but most of the analyses show oscillatory convergence. In the case of the torque of P1727 the convergence behaviour is not verified since the analysis exhibits oscillatory divergence. Relative errors are, however, comparable to those evidenced in the case of fully turbulent calculations, with the most problematic predictions for the P1727 test case. The apparent order of grid convergence in some cases is particularly high and this may explain the small estimated uncertainties. In any case, however, with the Medium mesh arrangement (approximatively between 5 and 6 Million cells, depending on the test case) the difference (the approximate relative error) with respect to the finer discretizations are sufficiently low using both turbulence models to accept this arrangement as appropriate for the complete set of calculations in reasonable computational times.

Fig. 13.

$L^{1}$ residuals convergence for the VP1304 test case.

Fig. 14.

Iterative convergence of $K_{T}$ and $10 K_{Q}$ for propeller VP1304 at $J = 1.2$ . Medium mesh arrangement.

An estimation of the iterative errors, for the VP1304 test case with the Medium mesh arrangement, is shown in Fig. 13 where the $L^{1}$ norm of the residuals was used for the analysis. An example of the iterative convergence is presented, for the same propeller, in Fig. 14. Fully turbulent analyses with the SST $k - ω$ model were carried out with $I_{% inlet} = 1 %$ and $μ_{T} / μ_{inlet} = 10$ . With the transition sensitive model, the inlet turbulence intensity and the turbulent viscosity ratio were set equal to 3% and 600 respectively, as from the results of the influence of inlet parameters on the turbulence intensity at the propeller plane proposed in next sections. A good convergence of residuals of velocity components and turbulent quantities to values below $10^{- 6}$ is achieved for both the turbulence models while, for pressure, residual values stagnate only just below $10^{- 4}$ . Similarly, the variation of propeller forces with the number of iterations shows a reasonable convergence achieved just after 3500 iterations in the case of calculations with the SST $k - ω$ model. Using the transition sensitive model reduces to 2500 the minimum number of iterations after convergence of forces. The standard deviation of thrust and torque, used as an estimator of the scattering of numerical calculations, is particularly low. In correspondence of the stagnation of the residual of pressure (between iteration 2500 and 3500 for fully turbulent calculations and between iteration 1000 and 2000 when using the transition sensitive model), standard deviations of both thrust and torque are about 0.15% and 0.55% of the average value respectively in the case of fully turbulent and transition enabled analyses. At convergence, in accordance with the good iterative convergence observed in Fig. 14, standard deviations drop at less than 0.03% and 0.05% respectively.

A similar behaviour of the iterative convergence was observed also for P1727 and P2772 test cases. In particular for the latter, a slightly higher scattering of thrust and torque values (standard deviation of about 1.7% of the average value) at convergence was observed as a consequence of the larger flow instabilities at the trailing edge discussed in the next. In the light of these results, the average of the last 500 iterations was used to collect propeller forces and moments for all the analyses proposed in the paper.

In the case of transition modelling, it is necessary to address the influence of the turbulence inlet quantities on the propeller performance. Results from the flat plate test case and from the preliminary investigations on predicted propeller trailing wakes at different turbulence intensities shown that the choice of inlet quantities is worth of special care. In a certain sense, for the specific problem of laminar to turbulent transition predictions, they can be seen as artificial (arbitrary) parameters for the tuning of the model, necessary to counteract the strong turbulent decay of the model, faster than in the case of the SST $k - ω$ . Since the inlet boundary is located four diameters upstream the propeller plane, a significant decay of turbulence quantities is indeed expected, determining an ambient turbulence intensity significantly different (lower) than that forced at inlet. In fully turbulent regime this has little influence on propeller forces, as already shown in Fig. 9 and, as long as the prediction of the propeller trailing wake is not an issue, any choice at inlet could be almost allowed. For the prediction of transition, instead, it would be necessary to have a reasonable (i.e. comparable to experiments) value of ambient turbulence (and turbulence length scale) on the test section by appropriately tuning the values at inlet, in order to capture as the flat plate test case in principle demonstrated, the actual transition point. For this aim, calculations with the reference inlet values for fully turbulent SST $k - ω$ analyses ( $I_{% inlet} = 1 %$ , $μ_{T} / μ_{inlet} = 10$ ) have been compared with three additional cases selected in order to vary the effective turbulence intensity on the propeller plane, achieved by changing the turbulence intensity and the turbulent viscosity ratio at inlet. Since, in a range coherent with similar calculations and experimental facilities data, only values of turbulence intensity on the test section were supposed for the analyses, in principle there are infinite possible couples of inlet turbulent quantities (k and ω or $I_{%}$ and $μ_{T} / μ$ ) satisfying a required turbulence intensity on the propeller plane. Without any, even more scarce, information on the turbulent length scale, from which an estimation of the specific rate of dissipation could be derived, the choice of inlet quantities arbitrary favours lower value of the eddy viscosity ratio with a non excessively high value of inlet turbulence intensity.

An example of the decay rate for the VP1304 propeller is shown in Fig. 15. As in the case of the T3A test case, the decay is faster and, due to the distance of the inlet from the propeller plane, it is difficult to achieve high values of turbulent intensity at the reference position also using very high values of turbulence and eddy viscosity ratio at inlet. On the other hand, as a consequence of the distance itself, the decay rate is very slow and across the region occupied by the propeller the turbulence intensity can be considered almost constant. As shown in the case of the flat plate, in any case, the variation of the turbulent intensity in the laminar region seems not significantly affecting the prediction of the transition location.

Additional calculations for each test case, in correspondence to sensibly higher Reynolds numbers achieved by increasing the rate of revolution, are included in the analyses. These outcomes, together with the resulting intensity levels on the propeller plane, are compared in Tables 9, 10 and 11 for the same advance coefficients (1.2, 0.5 and 0.5, respectively for the VP1304, the P1727 and the P2772 propeller) selected for the mesh sensitivity investigations. Comparisons of limiting streamlines on the three propellers blades are given, as well, in Figs 17, 19 and 21 for the design Reynolds numbers.

Fig. 15.

Decay of turbulence intensity in the computational domain as a function of inlet parameters and propeller rate of rotation. VP1304 test case.

Table 9

Sensitivity to turbulence parameters for the VP1304 test case at $J = 1.2$ . Pressure $(K_{T p}, 10 K_{Q p})$ and viscous $(K_{T v}, 10 K_{Q v})$ contributions to propeller $(K_{T}, 10 K_{Q})$ performance

	$I_{% inlet}$	$μ_{T} / μ_{inlet}$	$I_{% prop. plane}$	$K_{T} (K_{T p}, K_{T v})$	$10 K_{Q}$ ( $10 K_{Q p}$ , $10 K_{Q v}$ )
Fully turb. (a)	1%	10	0.38%	0.2588 (0.2677, −0.0089)	0.7263 (0.6770, 0.049)
Trans. (b)	1%	10	0.12%	0.2858 (0.2896, −0.0038)	0.7654 (0.7450, 0.021)
Trans. (c)	3%	600	0.94%	0.2881 (0.2918, −0.0038)	0.7717 (0.7512, 0.021)
Trans. (d)	5%	2000	1.70%	0.2858 (0.2897, −0.0039)	0.7660 (0.7450, 0.021)
Trans. (e)	10%	2000	1.75%	0.2861 (0.2901, −0.0039)	0.7667 (0.7457, 0.021)
Trans. (25 $rps$ ) (f)	7%	2000	1.38%	0.2933 (0.2968, −0.0035)	0.7784 (0.7589, 0.020)
Trans. (40 $rps$ ) (g)	10%	2000	1.07%	0.2985 (0.3021, −0.0036)	0.7907 (0.7705, 0.020)

Table 10

Sensitivity to turbulence parameters for the P1727 test case at $J = 0.5$ . Pressure $(K_{T p}, 10 K_{Q p})$ and viscous $(K_{T v}, 10 K_{Q v})$ contributions to propeller $(K_{T}, 10 K_{Q})$ performance

	$I_{% inlet}$	$μ_{T} / μ_{inlet}$	$I_{% prop. plane}$	$K_{T} (K_{T p}, K_{T v})$	$10 K_{Q}$ ( $10 K_{Q p}$ , $10 K_{Q v}$ )
Fully turb. (a)	1%	10	0.5%	0.1959 (0.1982, −0.0024)	0.2685 (0.2421, 0.0264)
Trans. (b)	1%	10	0.18%	0.2090 (0.2100, −0.0010)	0.2753 (0.2629, 0.0125)
Trans. (c)	3%	400	1.07%	0.2089 (0.2100, −0.0010)	0.2752 (0.2629, 0.0125)
Trans.(d)	5%	2000	2.28%	0.2087 (0.2098, −0.0010)	0.2749 (0.2625, 0.0125)
Trans. (e)	10%	2000	2.47%	0.2084 (0.2095, −0.0010)	0.2744 (0.2619, 0.0125)
Trans. (35 $rps$ ) (f)	5%	2000	1.73%	0.2119 (0.2127, −0.0008)	0.2745 (0.2646, 0.0099)
Trans. (50 $rps$ ) (g)	10%	2000	1.35%	0.2135 (0.2143, −0.0008)	0.2748 (0.2653, 0.0096)

Table 11

Sensitivity to turbulence parameters for the P2772 test case at $J = 0.5$ . Pressure $(K_{T p}, 10 K_{Q p})$ and viscous $(K_{T v}, 10 K_{Q v})$ contributions to propeller $(K_{T}, 10 K_{Q})$ performance

	$I_{% inlet}$	$μ_{T} / μ_{inlet}$	$I_{% prop. plane}$	$K_{T} (K_{T p}, K_{T v})$	$10 K_{Q}$ ( $10 K_{Q p}$ , $10 K_{Q v}$ )
Fully turb. (a)	1%	10	0.55%	0.1980 (0.2004, −0.0024)	0.2939 (0.2653, 0.0286)
Trans. (b)	1%	10	0.19%	0.2180 (0.2191, −0.0011)	0.3182 (0.3033, 0.0149)
Trans. (c)	3%	300	1.03%	0.2165 (0.2176, −0.0011)	0.3155 (0.3007, 0.0149)
Trans. (d)	5%	2000	2.48%	0.2194 (0.2205, −0.0011)	0.3200 (0.3051, 0.0149)
Trans. (e)	10%	2000	2.68%	0.2199 (0.2210, −0.0011)	0.3216 (0.3066. 0.0150)
Trans. (30 $rps$ ) (f)	5%	2000	1.86%	0.2288 (0.2296, −0.0008)	0.3209 (0.3104, 0.0105)
Trans. (50 $rps$ ) (g)	10%	2000	1.47%	0.2342 (0.2353, −0.0011)	0.3278 (0.3153, 0.0123)

The simultaneous comparison of propeller performance and skin friction coefficient distributions at different level of turbulence intensity reveals some critical issues of the laminar to turbulent transition predictions using the $k - k_{L} - ω$ turbulence model. With respect to fully turbulent calculations, the modelling of transition for these three test cases determines, for any choice of inlet quantities, a significant increase of both the thrust and the torque coefficients. This increase is higher (sometimes opposed) than that evidenced in similar calculations (but with the $γ - {Re}_{θ}$ ) on different test cases [5]. In those analyses, for instance, it was shown the role of pressure and friction on propeller forces and in particular the fact that the increase of torque due to the pressure when laminar flow is predicted is lower than its decrease due to the modification of the frictional forces. The total absorbed torque when laminar flow is accounted for in calculations was, then, reduced while it was possible to appreciate a slight increase of the delivered thrust consequent to the reduction of friction. Reduction of torque and increase of thrust is what has been observed, exactly using the $k - k_{L} - ω$ turbulence model and the OpenFOAM RANS library, in the case of the analyses of CLT and new Generation CLT propellers carried out in [21]. Instead, for the specific case of the VP1304, most of the results collected during the ITTC Benchmarking Test Case [24,33] confirm the trend observed in current calculations. All the analyses with any transition sensitive model proposed to the ITTC led to significantly higher values of both thrust and torque, the latter in contrast with the expected decrease of the frictional drag when laminar flow is present (as indicated in [5]). By contrast, other calculations [48] for the same geometry but with the application of a $γ - {Re}_{θ}$ transition based model showed a simultaneous decrease of the overestimation of both the thrust and the torque with respect to the values computed with fully turbulent boundary layer with the SST $k - ω$ approach.

The pressure distributions over the blade of Fig. 16 clarify the reasons of this increment. At the blade trailing edge, when the flow is predicted as laminar by using the transition sensitive model, it is possible to appreciate locally thicker pressure distribution, coherently to what observed for conventional and unconventional geometries by Shin and Andersen [2]. The contribution of the higher value of suction pressure on the back of the blade to propeller forces, in this specific case, is dominant with respect to the variation (reduction) of the skin friction coefficient.

Fig. 16.

Pressure coefficient distributions (non-dimensional with respect to $0.5 ρ N^{2} D^{2}$ ) on the VP1304 test case at the design advance coefficient $J = 1.2$ . Fully turbulent calculations compared to transition sensitive calculations with $I_{% inlet} = 3 %$ , $μ_{T} / μ_{inlet} = 600$ .

In the case of the P1727 propeller (and also for the P2772, for which no systematic data are available), results from the ITTC Benchmarking Test Case [25,33] are more scattered and it is not possible to assess the reliability of the trends (increment of both thrust and torque when the transition sensitive model is used) evidenced in Tables 10 and 11 based only on code comparisons. However, the development of laminar boundary layer over most of the blade discussed in the next paragraph suggests that the significant increase of propeller forces with respect to fully turbulent analyses, observed also for these two additional test cases, can be similarly ascribed to the higher suction pressure at the blade trailing edge.

By looking more in detail at the role of inlet quantities, differently to what observed for the simpler flat plate test case, a moderate change in the turbulence intensity when using the transition sensitive model seems not to significantly alter the computed propeller performance, and clear tendencies (decrease/increase of thrust/torque with increasing turbulence intensity) can not be identified either. A hint on the reasons of this behaviour can be found in the analysis of the skin friction distributions over the blades, which point out, for most of the cases under investigation, the non occurrence of transition from laminar to turbulent flow regardless the ambient turbulence intensity. For any choice of the inlet quantities, calculations with transition, indeed, show almost all the blade surface affected by laminar flow, identifiable by the lower values of friction and by the significantly different orientation of the limiting streamlines compared to fully turbulent analyses even if all the three propellers are working in critical Reynolds number conditions. When transition occurs, indeed, the slope of streamlines changes noticeably: they are mainly aligned with the chordwise direction in fully turbulent conditions while in the case of laminar boundary layer they have a significant radial component, as observed using paint tests [5,8]. Also if specific visualizations of limiting streamlines for these three propellers under investigation are not available, results of present analyses resemble the usual streamlines orientation and change depending on the flow regime. Streamlines originating from the leading edge have a prevalent chordwise direction, indicating well-defined leading edge attached flows using both the turbulence models. In fully turbulent conditions, with the SST $k - ω$ , the tangential direction of streamlines is confirmed also moving to the trailing edge, except close to the blade root where separation may occur. Using the transition sensitive model, as the streamlines approach the trailing edge, they become more radially oriented and, without the occurrence of transition to turbulent flows, they maintain, as observable for any test cases and choice of turbulence values of Figs 17, 19 and 21, this orientation over almost all the blade.

Fig. 17.

Limiting streamlines and skin friction coefficient distribution (non-dimensional with respect to $N^{2} D^{2}$ ) on the VP1304 test case (suction side) at the design advance coefficient $J = 1.2$ for different values of turbulence intensity at the propeller plane.

Fig. 18.

Limiting streamlines and skin friction coefficient distribution (non-dimensional with respect to $N^{2} D^{2}$ ) on the VP1304 test case (suction side) at the design advance coefficient $J = 1.2$ . Higher Reynolds number cases. Comparison with results from [48,69].

Fig. 19.

Limiting streamlines and skin friction coefficient distribution (non-dimensional with respect to $N^{2} D^{2}$ ) on the P1727 test case (suction side) at the advance coefficient $J = 0.5$ for different values of turbulence intensity at the propeller plane.

Fig. 20.

Limiting streamlines and skin friction coefficient distribution (non-dimensional with respect to $N^{2} D^{2}$ ) on the P1727 test case (suction side) at the advance coefficient $J = 0.5$ . Higher Reynolds number cases. Comparison with results from [48].

If compared with similar calculations with other transition sensitive turbulence models [5,48] current analyses with the $k - k_{L} - ω$ seem, for these complex three-dimensional geometries, less prone to transition, even if the ambient conditions (i.e. sufficiently high turbulence intensity at the propeller plane achieved through higher turbulence intensity at inlet and increased eddy viscosity ratios) are favourable (and even higher to those proposed in [5]) to the development of turbulent flows.

In the specific case of the VP1304 and of the P1727 test cases, more detailed comparisons are possible with the calculations proposed in [69] and in [48]. For the VP1304 propeller, differences are significant also if current results are compared to the calculations without the inclusion of the crossflow correction [35], which was specifically developed to anticipate the laminar to turbulent transition in the presence of secondary flows like those at the tip of the blade.

Both calculations, when transition is enabled, show radially oriented streamlines at the root of the blade, with a convergence and an accumulation close to the trailing edge where separation occurs and banded distribution of skin friction coefficient as a result of flow instabilities. In the outer part of the blade, current calculations still show radially oriented streamlines and then mainly laminar flow, identifiable by the smooth distribution of skin friction in the radial direction. By contrast, the results of [69] and [48] using the $γ - {Re}_{θ}$ model clearly identify turbulent flow conditions, highlighted by the higher values of the skin friction, by its strong gradient along the radial direction and by the abrupt variation of the streamlines orientation, from radial to chordwise. With the $k - k_{l} - ω$ model these features are postponed: they are predicted only for the significantly higher chord-based Reynolds numbers of Fig. 18 (from $0.86 \cdot 10^{6}$ to $1.43 \cdot 10^{6}$ and $2.30 \cdot 10^{6}$ , achieved increasing the propeller rate or revolution from 18 respectively to 25 and 40) and a sufficiently high values of the ambient turbulence intensity. In these case, the $k - k_{l} - ω$ model predicts as well turbulent flow on the outer portion of the blade, from $r / R = 0.75$ to tip (at the highest Reynolds number considered) and from midchord to the trailing edge as clearly pointed out by the sudden increase of skin friction and the local tangential direction of streamlines. Simultaneously, as the flow tends to the turbulent regime, a progressive reduction of the skin friction streaks observed at the trailing edge can be appreciated too. Similar streamlines arrangements and friction coefficient oscillations at the blade trailing edge have been observed also using the $γ - {Re}_{θ}$ model [2,48] and also in those cases a progressive reduction of the skin friction streaks was achieved increasing the Reynolds number. These results and this postponing tendency, with transition to turbulence unrealistically taking place at a Reynolds number significantly higher than the critical one, agree moreover with the calculations (with the same transition model) of [68], which showed a significant portion of the outer part of the blade subjected to turbulent boundary layer only for Reynolds numbers of the order of $5 \cdot 10^{6}$ .

On the contrary, the comparison between the two turbulence models ( $γ - {Re}_{θ}$ from [48] and $k - k_{l} - ω$ of present analyses) is better in agreement when the P1727 propeller is considered. For this test case, indeed, the predominant laminar nature of the flow is similarly predicted: streamlines are radially aligned over most of the blade and instabilities are clearly seen at the trailing edge, regardless the transition sensitive turbulence model and the local mesh arrangement. Also an increase of the chord-based Reynolds number (from $0.47 \cdot 10^{6}$ at 18 rps to $0.926 \cdot 10^{6}$ at 35 $rps$ ) is not sufficient for any of the transition models applied, the current one and the $γ - {Re}_{θ}$ of [48] to stimulate the transition to turbulent boundary layer (which is always restricted to a small region at the trailing edge) over a significant portion of the blade, as shown in Fig. 20. With the $k - k_{l} - ω$ model of present calculations, only at 50 $rps$ (Reynolds number of $1.32 \cdot 10^{6}$ , Fig. 20(c)) there is a hint of turbulent flow at the outer radii, together with a progressive reduction, already observed for VP1304, of the banded arrangement of streamlines.

Results for the P2772 propeller are, qualitatively, very close to those observed for the P1727, with the working points being very similar (i.e. model scale Reynolds number and thrust coefficient at the same considered advance coefficient). Also for this propeller the application of the $k - k_{l} - ω$ model predicts laminar flow over most of the blade regardless of the relatively wide range of ambient turbulence intensity at the propeller plane accounted in the analysis and, exactly as in the case of P1727, also a moderate increase of the chord-based Reynolds number (up to $0.99 \cdot 10^{6}$ ) by varying the propeller rate of revolution is ineffective to stimulate turbulent transition and a considerable portion of turbulent flow over the blade (Fig. 22). An appreciable extension of the turbulent boundary layer over the propeller blade, from midchord to the trailing edge at the outermost radii, appears only for even higher Reynolds number ( $1.66 \cdot 10^{6}$ ) possible at a propeller rate of revolution equal to 50.

In addition, an increase of the propeller rate of revolution (i.e. more extended turbulent regions), at least for the two higher Reynolds numbers under investigation, does not correspond to propeller performance predicted with the transition sensitive $k - k_{l} - ω$ model converging towards the fully turbulent calculations. This non-expected trend can observed in Tables 9, 10 and 11 for all the three test cases. It is worth to note, however, that also fully turbulent analyses with the SST $k - ω$ suffer a certain influence of increasing the propeller rate of revolution, in particular for the VP1304 which exhibits at the highest rate of rotation (40 rps) an increase of both thrust and torque up to 3% with respect to the results at the reference working conditions.

Fig. 21.

Limiting streamlines and skin friction coefficient distribution (non-dimensional with respect to $N^{2} D^{2}$ ) on the P2772 test case (suction side) at the advance coefficient $J = 0.5$ for different values of turbulence intensity at the propeller plane.

Fig. 22.

5.3. Propeller performance under transition conditions: Open water diagrams

Without paint tests to validate the transition location and the relatively low influence of inlet quantities on transition occurrence, the choice of the most appropriate parameters to predict the open water propeller performance subjected to the influence of transition phenomena is not clear. The obvious choice would be to set up simulations (i.e. by changing the inlet turbulence intensity and eddy viscosity ratio) to have the ambient turbulence intensity in correspondence to the propeller equal to that measured during experiments. The availability of this information, however, is not granted and in some cases [5] the ambient turbulence intensity was calibrated (i.e. considering it as tuning parameter of the model) in order to have a good agreement with experiments rather than fixing it to measured values. Since the turbulence intensities (nor the turbulent length scale) in the towing tanks were not confirmed, present calculations were carried out using typical values for this kind of experiments in the towing tank [41]. For all the three test cases, the ambient turbulence intensity at the propeller plane (computed at the design advance coefficient) was set to about 1%, which is a reasonable value for the water at rest in the towing tank (besides being already used for similar calculations and with the same turbulence model [68]), even if at the critical Reynolds numbers of the measurements a sensible overestimation of the laminar boundary layer by the $k - k_{l} - ω$ model is expected for some geometries (in particular the VP1304, as from the previous discussion). This value, without any additional information on the turbulent length scale, was obtained by arbitrary setting an inlet turbulence of 3% and by tuning the eddy viscosity ratio to 600, 400 and 300 respectively (case “c” of Tables 9, 10 and 11). Of course this value in principle is not independent by the inflow velocity (i.e. advance coefficient) but, in the light of the scarce sensitivity demonstrated by the $k - k_{l} - ω$ model, it has not changed for calculations at advance coefficients different from the design ones.

Fig. 23.

Open water propeller performance for VP1304 and P1727 Tip test cases. Comparison with measurements and data (envelopes of minimum and maximum values) from the ITTC Benchmarking Test Case [33].

The comparison between the numerical predictions (both fully turbulent with the SST $k - ω$ and transition sensitive analyses using the $k - k_{l} - ω$ model) and the available measurements from open water tests is presented in Figs 23 and 24. In the case of VP1304 and P1727 also numerical results (envelopes of minimum and maximum predicted values) from the ITTC Benchmarking Test Case are included. As already observed in the case of the sensitivity analyses carried out to assess the role of turbulent quantities on the predicted flow, a considerable effect of the application of the transition model, which lead to significantly improved results with respect to fully turbulent analyses, can be appreciated. Its influence is particularly positive for moderate (close to design) and high advance coefficient.

Fig. 24.

Open water propeller performance for P2772 test case.

In the case of the VP1304 propeller, when using the transition sensitive model, the thrust errors are in the order of 1.2% for $J = 0.6$ to −2.7% for $J = 1.2$ which are significantly lower than those observed with the application of fully turbulent analyses, when the underestimation of propeller performance is up to −12.5% at the design advance coefficient. Similar torque errors are obtained: when transition is included, discrepancies are of about 1%, with a maximum underestimation at the design advance of −3.4% which is a substantial improvement with respect to the significant underestimation (−6.4%) predicted when turbulent flow is assumed. Predictions for P1727 and P2772 are similarly improved by the application of the transition model. For the P1727, in particular, the error at the design advance coefficient is nullified (versus an underestimation of −6.3% and −2.4% respectively for thrust and torque when using the SST $k - ω$ ) and the overall difference with respect to measurements is significantly reduced with respect to usual fully turbulent analyses. At $J = 0.9$ relative errors are still high but it depends by the fact that thrust is near zero. In absolute values, however, the use of the transition sensitive model significantly reduces the differences if compared to the SST $k - ω$ . Only at the lower advance coefficients both thrust and torque are slightly overestimated (about +3%) and calculations with the turbulent assumption provide better agreement with experiments (−0.2% for thrust, +1.7% for torque). For the P2772 test case, the overestimation of propeller forces at lower advance coefficient when using the $k - k_{l} - ω$ model corresponds also to a different slope of the performance curves. As pointed out in [69] for the VP1304 test case and in [8] in the case of ducted propellers, the loading condition of the blade has, especially for the suction side flow, a certain influence on transition, fostering at lower advance coefficients the development of turbulent boundary layer. This could be related to the role of the cross flow seen in [48], obviously higher at higher loadings. Together with the tendency of postponing transition phenomena and with low sensitivity (at the tested Reynolds numbers) of the proposed turbulence modelling to the ambient turbulence intensity (lower advance speeds, at unvaried inlet turbulence intensity, lead to higher turbulence intensity to the propeller), the observed overestimation of laminar flow over the propeller blades could explain the resulting overestimation of propeller forces with respect to fully turbulent calculations.

No information about the input uncertainties of the parameters (the turbulence intensity during measurements [33] is not provided at all) is available as no uncertainties estimation of the measurement are provided in the ITTC database [24,25]. The procedure for validation proposed by ASME [4], then, can be used comparing the difference between predicted and measured values, summarized for the advance coefficient of the grid convergence studies in Table 12, only with the numerical uncertainties simply based on the Grid Convergence Indexes with a safety factor [59] (currently 1.25) of Tables 6, 7 and 8.

Table 12

Comparison between numerical and experimental results

	Turbulent		Transition

	$Δ K_{T} %$	$Δ 10 K_{Q} %$	$Δ K_{T} %$	$Δ 10 K_{Q} %$
VP1304, $J = 1.2$	−12.27%	−6.40%	−2.34%	−0.55%
P1727, $J = 0.5$	−6.27%	−2.36%	−0.05%	0.07%
P2772, $J = 0.5$	−11.52%	−8.82%	−3.25%	−2.11%

In none of the case, with the exception of the P1727 using the transition sensitive turbulence model (which, however, shows some convergence issues) the comparison error is lower than the (very low) numerical uncertainties. Comparing fully turbulent with transition sensitive calculations shows, obviously, a reduction of the error associated to modelling in the case of transition enabled calculations, since the fully turbulent model miss the prediction of relevant phenomena, like transition, for propeller performance estimation. Even if based on the procedure of [4] validation is not achieved, similarly to what obtained [5] for different propeller geometries and different codes, neither when transition models are used, the resulting assessment of the modelling error (between 2 and 3%) of the transition sensitive calculations can be considered satisfactory for application purposes. The validation uncertainty (i.e. without any estimation of the uncertainty of experiments, the numerical uncertainty alone) of these test cases is, however, particularly low, probably as a consequence of unreasonable large values of the order of grid convergence and to a non-optimal selection of the grids for the Richardson extrapolation. More robust estimations of the numerical uncertainties for such complex phenomena and geometries, considering for instance more than three computational grids [14], could in the end provide a more reliable verification and validation process.

6. Conclusions

The goal of the present work was to verify the possible improvements of model scale propeller performance predictions using transition sensitive turbulence models and, in particular, the $k - k_{l} - ω$ model of the OpenFOAM library. The simultaneous occurrence of laminar and turbulent flow can have a significant impact on computed model scale propellers performance: extending and validating the ability to predict transition flows would obviously provide useful tool for design and analysis of complex and more efficient propulsive solutions.

Three test cases from acknowledged benchmarks and research projects have been proposed to assess the reliability and the impact of the transition sensitive turbulence model for the prediction of model scale propeller performance: two conventional (VP1304 and P2772) propellers and one non-conventional, tip loaded, geometry (P1727) for all of which dedicated towing tank tests at low Reynolds numbers suitable for laminar to turbulent transition were available. Numerical analyses have been carried out also using the standard, fully turbulent, SST $k - ω$ model to further stress the issues related to usual analyses with the fully turbulent boundary layer assumption.

A preliminary analysis of the features of the transition sensitive turbulence model adopted for propeller analysis was carried out considering the well-established T3A case from the flat plate experiments database provided by the European Research Consortium on Flow, Turbulence and Combustion (ERCOFTAC). In the case of the simple, well-controlled geometry proposed by the Consortium, the $k - k_{l} - ω$ model was able to reasonably predict the transition location once the turbulent inlet parameters (turbulent intensity and turbulent viscosity ratio) were tuned in order to have, at the flat plate leading edge, the same turbulence intensity measured during experiments. On the basis of the proposed calculations, moreover, the local decay rate of the turbulence intensity along the laminar part of the boundary layer seems less influential on the transition location. These calculations suggested the need of a proper tuning (sometimes artificial and unrealistic) of the turbulent quantities at the computational domain inlet also in the case of the propeller calculations to counteract the turbulent decay and achieve the appropriate amount of turbulence intensity on the reference location represented, usually, by the propeller plane.

The successive comparison with propeller model tests shows the substantial improvements achievable by including the role of laminar boundary layer also in the prediction of propellers flow and relative performance. For all the cases, the application of the transition sensitive model with a reference turbulent intensity at the propeller plane of about 1% (as from usual towing tank measurements) provides a very good (better) agreement with experimental values, in the range of 1% to 3% for VP1304 and P1727 and between 2% and 8% for P2772 depending on the advance coefficient. These are differences substantially lower than those observed in the case of calculations with the fully turbulent model (up to 25% in lightly loaded conditions, with an average discrepancy of 8% at the design point). For all the three cases, the application of the $k - k_{l} - ω$ model yields predicted values of both thrust and torque higher than those using the fully turbulent model. This is not in accordance with the commonly expected influence of a reduction of friction due to laminar boundary layer, which intuitively should result into a slight increase of thrust and a more substantial reduction of torque. This opposite tendency shown by the proposed transition sensitive calculations, however, is confirmed by the collection of results of the ITTC Benchmarking Test Case for the VP1304 and the P1727 propellers [33] and by similar calculations in the case of other unconventional propellers [21]. In this regard, the analysis of the pressure distributions over the blades demonstrates the influence of developed laminar boundary layers on pressure, showing the reasons of the higher predicted forces: higher adverse pressure gradient and higher value of suction close to the trailing edge of the back of the blade when the transition sensitive model is employed [2].

If, from a pure performance point of view, the improvement ascribable to the transition modelling are significant, the analysis of the skin friction distributions and of the limiting streamlines (in the end, of the occurrence of transition) show the critical points of the current modelling approach. Differently to what observed for the flat plate test case, the transition sensitive model is substantially less sensitive to turbulence intensity when applied to complex three-dimensional geometry subjected to adverse pressure gradients and significant cross flow. At the Reynolds numbers of the experiments, any choice of turbulence intensity at the propeller plane leads to differences in the computed thrust and torque coefficients barely appreciable since the blades only show a strong laminar flow, without any transition to turbulent, which is doubtful especially for the VP1304 model at its functioning point. Compared to similar analyses with different turbulence transition modelling [48,69], the differences are clear in the case of the VP1304 propeller, where the ineffectiveness of current calculations to capture the transition to turbulent boundary layer in the outer part of the blade close to tip is evident. For P1727 (but similar considerations may hold in the case of the P2772 due to the similar functioning conditions of these two propellers), instead, the outcomes from current analyses better agree with similar calculations available in literature [48], which show also by using the $γ - {Re}_{θ}$ model, a blade completely covered by laminar boundary layer at the design Reynolds number.

In the end, current analyses prove an unrealistic tendency of the $k - k_{l} - ω$ model towards postponement of the transition phenomena on the propeller blades, where a significant adverse pressure gradient may exist and then confirming the outcomes of previous analyses [16,66]. Only at higher Reynolds numbers together with sufficiently high values of ambient turbulence sustained by high eddy viscosity ratios, indeed, calculations with the $k - k_{l} - ω$ model predict transition on the suction side, with a qualitatively behaviour (orientation of streamlines and variation of skin friction distribution) in agreement with similar calculations and relevant paint tests. This is particularly true in the case of the VP1304 which shows a hint of turbulent boundary layer when the Reynolds number is raised to $1.43 \cdot 10^{6}$ and a well developed turbulent flow with streamline chordwisely oriented in the outer part of the blade at a Reynolds number of $2.30 \cdot 10^{6}$ . For the other two test cases this tendency is confirmed even if the size of the turbulent boundary regions is obviously smaller, since also at the higher propeller rate of rotations (up to 50 $rps$ ) selected for current analyses, the chord-based Reynolds is compatible with a predominant laminar boundary layer over the blades. Based on these tendencies, the analysis of the influence of inlet turbulent parameters proposed for the propeller test cases as a consequence of the successful tuning for the T3A flat plate case, is less significant. Further analyses are then necessary using higher (even if unrealistic) Reynolds numbers to stimulate the transition and verify the role (and the consequent opportunity of tuning) of the inlet parameters on boundary layer development.

Additional, and careful, analyses are finally needed in order to investigate, also by comparing results from the $γ - {Re}_{θ}$ model available in the last release of the OpenFOAM library [63], the reasons of these discrepancies through systematic analyses at varying Reynolds numbers and different loading conditions. The role of cross flow, especially at higher loading, could be significant in fostering transition. Worth of investigation could be, as well, the calibration, on the basis of detailed experimental investigations of transition of flows over complex geometries like propeller blades, of (some of) the constants of the $k - k_{l} - ω$ model which drive bypass and natural transition to improve the laminar to turbulent transition prediction at realistic critical Reynolds numbers.

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Improving model scale propeller performance prediction using the k − k L − ω transition model in OpenFOAM

Abstract

Keywords

1. Introduction

2. Test cases: VP1304, P1727 and P2772 propellers

3.1. RANS equations

3.2. The SST k − ω turbulence model

3.3. The k − k L − ω transition model

4. The ERCOFTAC T3A test case

References

3.2. The SST $k - ω$ turbulence model

3.3. The $k - k_{L} - ω$ transition model