J o u r n a l o f S h i p Research, V o l . 6 0 , N o . 2, J u n e 2 0 1 6 , p p . 7 8 - 9 1 h t t p : / / d x . d o i . O r g / 1 0 . 5 9 5 7 / J O S R . 6 0 . 2 . 1 5 0 0 1 0
Net and Gross Thrust in Waterjet Propulsion
Arash Eslamdoost, Lars Larsson, and Rickard BensowDepartment of Shipping and Marine Technology, Ctialmers University of Technology, Gothenburg, Sweden
The measurement of the net thrust of a waterjet unit is a cumbersome task and thus as an alternative, the thrust of the waterjet system is expressed based on the momen-tum flux change through the waterjet unit, called gross thrust. The relation between net thrust and gross thrust is not fully understood, and in the current paper this relation is investigated by employing numerical simulations of the flow around a planing waterjet-propelled hull. The validation of the bare hull and self-waterjet-propelled hull simulations is carried out through comparison of the computed results with experimental data. Keywords: waterjet propulsion; thrust deduction; net thrust; gross thrust; intake drag; exit drag
1. Introduction
A L T H O U G H T H E principal mechanism o f a vvaterjet-propulsion system is very mucii the same as a conventional propeller, the degree o f integration with the hull is different, making it hard to use the same concepts for studying the thrust and powering o f these systems. The net thrust o f the propeller can be obtained by measuring the force transmitted through its shaft, but since there is not just a single contact point between the waterjet unit and the hull, the net thrust measurement cannot be easily accom-plished f o r the waterjet unit. Instead, another thrust force, which is simpler to measure, is defined to express the magnitude o f the waterjet unit thrust. The new thrust is called gross thrust and is obtained by the measurement o f the momentum f l u x change through the waterjet control volume.
There have been some efforts to investigate the waterjet-hull interaction and to estimate the power o f the wateijet system. Etter et al. (1980) discuss the differences between the model tests o f the waterjet-driven craft and the propeller-driven one. They try to define propulsion terms and experimental procedures f o r evalua-tion of the waterjet and hull performance together f r o m tests of separate components. Also, analogies between their proposed method f o r waterjets are noted in comparison to conventional propellers. Dyne and Lindell (1994) question the method o f obtaining the required net thrust f r o m a thrust deduction fraction and instead introduce a direct method giving the shaft power without using any propulsive factor. Coop (1995), in his doctoral
IVtanuscript received at SNAME headquarters January 22, 2016; revised manuscript received February 13, 2016.
thesis, investigates the interaction between wateijet and hull using model-scale and full-scale measurements as well as empiiical and analytical methods, which are based on the Savitsky planing per-formance equations, to understand the possible mechanisms con-tributing to the overall interaction effect. A review of the methods for obtaining the nozzle momentum flux is presented in this thesis. Among these methods, a Prandtl rake is used for measuring the nozzle momentum flux in this shidy.
The report o f the Intemational Towing Tank Conference ( f l T C ) Specialist Committee on Waterjets (PITC 1996) discusses possible power prediction methods for wateijet propulsion systems. Eventu-ally, two different methods are proposed: First, the calculation o f the thrust force from the momentum flux and second, the direct measurement of thrust. A method for calculating the gross thmst of a wateijet system is presented. Moreover, intemal intake losses and scaHng effects are also discussed. I n his thesis work, van Terwisga (1996) indicates that a difference between gross thrust and net thrust may occur especially ai-ound ship speeds where the transom is cleared. This difference is practically zero for higher speeds and therefore, the difference between gross thnist and bare hull resistance is a good measure of the resistance increment of the hull due to the waterjet-induced flow. The I T T C Specialist Com-mittee on Vahdation o f Waterjet Test Procedures (fTTC 2002) tries to defme a standard test procedure for wateijet systems. General comments are provided on the design procedure o f wateijet ele-ments, such as intake and pump, and the procedure to investigate the waterjet-huU interaction. In order to calculate the momentum and energy flux, a different formulation is introduced compared to the formulation adopted by the I T T C Specialist Committee on Waterjets ( l l t C 1996). Further discussions of the method presented
by this ITTC Committee are canied out by the 24"' ITTC Speciahst Committee on Waterjets (ITTC 2005).
Bulten (2006), in his doctoral thesis , studies the dynamic forces on the waterjet unit without considering the interaction with the hull. I n contrast with van Terwisga (1996), he concludes that the net thrust and gross thmst are significantly different. These con-tradictory conclusions might result in different thiiist deduction values. The estimation of the wateijet system powering has been the focus of several more researchers (Cusanelli et al. 2007; Jess up et al. 2008; Delaney et al. 2009; Kandasamy et al. 2010; Takai et al. 2011; Daniele et al. 2012); but, to the knowledge of the authors of this paper, there has been no further detailed study of the difference between the net thrust and the gross thrust of waterjet systems since the work o f van Terwisga (1996) and Bulten (2006). The main focus of the current paper is to f i l l this knowledge gap and understand the parameters contributing to the differences between the net thrust and the gross thrust.
In the following, first, the definitions required for the analysis of the wateijet system are introduced in Section 2. Then, in Section 3, the self-propulsion computation method is presented and the results of this method are compared with the measured values. Finally, in Section 4, the relation between the net thrust and the gross thrust is investigated.
2. Theory
2.1. Surfaces and control volume
Figure 1 shows the cross section of a waterjet propulsion unit and the control volume (enclosed by surfaces 1, 2, 6, and 8), which is normally applied for the system analysis. The numbering of the surfaces shown in Fig. 1 is the same as those introduced by van Terwisga (1996). Suiface 1 is named the capture area and is located far enough in front of the intake ramp tangency point before inlet losses occur. As a practical solution, the I T T C Spe-cialist Committee on Validation of Waterjet Test Procedure rec-ommends one inlet length forward o f the ramp tangency point (ITTC 2005). The dividing streamtube is shown as surface 2 in Fig. 1. This streamtube is an imaginary surface, which separates the flow drawn into the ducting system f r o m the rest o f the flow field and no flow crosses this surface. Surface 4 is the part of the ducting channel, which lies outside the streamtube. Thus, it belongs neither to the waterjet control volume nor to the huU sur-face. It is enclosed by the stagnation hne on the intake lip (cut water), and the line where the channel merges with the flat bottom. The outer border is fixed by the geometry of the channeFhuU junc-tion, whereas the inner border of surface 4 is speed dependent and varies f r o m one operating condition to another. The waterjet system internal material boundaries are indicated as surface 6.
Suiface 7 is the boundary area of the pump control volume and surface 8 represents the nozzle discharge area. I f the nozzle dis-charge is designed such that all the outgoing streamlines are parallel with the nozzle-exit diiection, the pressure at the nozzle exit will be atmospheiic. Otherwise, the atmospheric pressure w i l l occur at another section downstream of the nozzle exit. The sec-tion at which the jet reaches its minimum diameter is called vena contracta and is marked as surface 9 in Fig, 1,
2.2. Thrust and thrust deduction
Having defined the control volume, and considering the coor-dinate system, x, y, and z, to be Cartesian and earth fixed (as shown in Fig, 1), the thrust force exerted on the control volume may be obtained by applying the momentum conservation law over the control volume of which the flow enters through A [ and exits through A^. The change in momentum flux over the waterjet system control volume is equal to the sum of the forces acting on this control volume.
pUi{ittni,)dA aijiijjdA + Ai+Ai+At+As
(i)
where the left-hand side is the net momentum flux through the control volume. The Einstein notation is used in this equation, in which i,J, and k range over 1, 2, and 3. The normal v e c t o r p o i n t s out of the control volume; hence, when the velocity vector, ii, exits the control volume, the product of x n is positive and when the velocity vector is entering the control volume, this product becomes negative. The first term on the right-hand side is the sum of the external forces acting on surface of the control volume and the second tenn represents the pump force acting on fluid. The last term on the right-hand side is the body force acting on the confrol volume.
The external forces acting on the control volume are represented by the tensor ay. This tensor can be split into two parts as follows:
= -pÖij + Tij (2) where p and Xy are the time-averaged pressure and sheai^ stress ten-sors, respectively, and 5,y, the Kronecker delta defined as follows:
5;; 1 It I = j
0 i f i ^ j - (3) The gross thrust, f g , is defined as "the force vector pertinent to the change in momentum flux over the selected control volume, acting on its environment" (van Terwisga 1996). This is basically the definition of the term on the left-hand side of equation (1). The gross thrust is a force vector but since the horizontal compo-nent of this vector is of main interest, shortly, this compocompo-nent of the gross thrust, Tg^y, is called the gross thrust, Tg.
T, pu.,{iU-iu)dA. A,+As
(4)
Fig. 1 Section cut through the waterjet ducting system
'The r r r C specialist Committee on Validation of Waterjet Test Procedures (ITTC 2005) suggests using Ag instead of A^. The reason for not using this suggested control volume is discussed in Section 2.3.
Since tlie gross tlirust is the reaction force exerted by the control volume on its environment, the minus sign makes the gross thrust point in the forward direction.
Considering the material boundaries of the waterjet system, another thrust force may be defined. "The net thrust, defined as the force vector acting upon the material boundaries of the wateijet system, directly passing the force through to the h u l l " (van Terwisga 1996). From now on, the horizontal compo-nent of the net thrust vector w i l l be called the net thrust, T„^^, which is defined as.
(5)
where is the horizontal component of the external forces acting on the control volume (a^ = Oijiij).
It should be noted that the thi-ust deduction fraction, /, is not the same as the conventional thrust deduction fraction, t, employed in propeller/hull interaction theory. For a conventional propeller, / relates the resistance of the bare hull, Rbh, to the net thrust required for driving the hull at a certain Froude number. How-ever, f o r a waterjet-driven hull, the thrust deduction fraction relates the resistance of the bare hull to the gross thrust and is defined as follows:
(6)
Using the ratio of the bare hull resistance and the net thrust of the waterjet system, it is possible to define a fraction which is similar to that employed i n conventional propeller/hull theory. It reads as follows:
i net
(7)
Basically, the net thrust o f a wateijet unit is equal to the resis-tance of the self-propelled huU and, therefore, the resisresis-tance incre-ment fraction, as it is obvious f r o m its name, reflects the change in the resistance of the wateijet-propelled hull compared to the bare hull resistance.
2.2.1. Relation between the net thrust and the gross thrust. Employing the definitions for the gross thnist and net thnist, equa-tions (4) and (5), the a-component of equation (1) may be rewritten as follows:
0^d^ -h üydA. (8) Ai+Ai+Ai
The right-hand side of equation (8) is the sum o f the intake, Di, and the exit drag, Dg, which are defined as follows:
cy.vd^ - I - o.vd/l, A\+A2 A4
a^dA.
(9)
(10)
The intake drag is the resultant of the forces acting on the capture area (Ai), the streamtube which is part of the control volume for deteiTnining gross thrust (/I2), and the part of the ducting channel that is outside the streamtube ( . ^ 4 ) . The exit drag is the force acting on the nozzle exit area (^g).
In order to relate the net thmst to the gross thrust, a jet thmst deduction fraction is defined as follows:
(11)
By applying the definitions of the net thrust and gross thmst i n equation (11), tj reads as follows:
a^dA A,+A2+As
a,ydA (12)
The jet thmst deduction fraction can be obtained using either one of equation (11) or (12). Equation (11) is based on the difference between two large numbers: the gross thmst and the net thmst, while equation (12) contains the sum of small forces acting on Al, A2, A4, and A^.
The combination of equations (11) and (7) yields: r g ( l - 0 ) ( l - O = ^ b h . Combining equations (13) and (6) yields:
t = tr + t j - t , X t j .
(13)
(14) In the literature, which has been published so far, it has been stated lhat tj value is negligibly small except at the transom clear-ance Froude number and, therefore, the second-order terms on the right-hand side of equation (14) can be neglected (van Terwisga 1996; r i T C 2005; Eslamdoost et al. 2014). I n Section 4.4, it w i l l be shown that may be large when the nozzle exit is not venti-lated and therefore the second-order term in equation (14) cannot be neglected.
2.3. IModification to the ITTC recommended procedure With the intake and the exit drag defined, it is possible to com-pare the recommended formulations by i r T C 1996 and f T T C 2005 with the ones used in this paper for obtaining the gross thmst of a wateijet system. As indicated earher, the control volume suggested by the I T I C Specialist Committee on Validation of Waterjet Test Procedures (ITTC 2005) is defined such that the f l o w exits the control volume thi'OUgh Ag rather than A^. I f the jet is discharged into the air, the pressure at the vena contracta {A>)) w i l l be atmo-spheric and there w i l l be no exit drag i f A9 is used as the exit of the control volume. But i f the nozzle exit is not ventilated, the pressure at the vena contracta w i l l not be atmospheric and an exit drag w i l l emerge. The measurement of the position of the vena contracta is not an easy task and it is much more complicated when the nozzle exit is not ventilated. On the other hand, the contribution of the pressure force acting on the capture area ( ^ 1 ) is not dnectly taken into account when obtainmg the gross thmst and instead this contribution is treated as intake drag. Including the contribution of the pressure force at the exit and excluding it at the capture area does not seem consistent. This inconsistency did not
exist in tlie eai'lier fecommended procedufe by tlie ITTC Specialist Committee on Wateijets (ITTC 1996). According to that proce-dure, the vena contracta is used as the exit of the control volume but then the pressure force acting on the capture area is also taken into account in the definition of the gross thrust. So, it would be more consistent to use either the suggested procedure by the ITTC Specialist Committee on Waterjets (ITTC 1996) or the waterjet control volume employed in this paper and treat the pressure force contribution on the capture area and the nozzle exit as the intake and exit drags.
3. Self-propulsion computations and validation
In this section, first, the method used for the numerical simula-tion of wateijet hull self-propulsion is presented, followed by a verification of the numerical method. This means determining the numerical eiTor obtained f r o m a systematic grid refinement study. Then, the measurement technique used at SSPA Sweden A B in Gothenburg for the self-propulsion tests is discussed briefly. The final part is the validation against experiments. This pait also contains the steps carried out for obtaining the gross thrust in the numerical simulations.
3.1. Numerical method
The numerical simulations for the bare hull were carried out with the code STAR-CCM-|- 8.02 and are based on the Reynolds-Averaged Navier-Stokes (RANS) equations. A finite volume method in combination with control volumes consisting of arbitrary poly-hedral volumes is used in this code to solve the unsteady mass and momentum conservation equations in integral form. A n implicit unsteady time stepping method is used. This method has a wide stability range (Courant number greater than 1) and therefore allows large local time step. The Volume of Fluid (VOF) method is used to obtain the volume fraction of the liquid, which adds one more equation to the system of equations. Convective terms in this equa-tion are discretized using the High-Resoluequa-tion Interface Capair-ing (HRIC) scheme (Muzaferija and Peric 1999). The free-surface inteiface is expected to be sharp since this scheme resolves the free-surface within typically one cell. The realizable k-e turbu-lence model is used to solve the turbuturbu-lence effect on the mean flow. Hence, two more equations need to be solved, one for the kinetic energy, k, and one for the dissipation, e. A wall func-tion model f o r high y'^ values (y'^ > 30) is used and i f the mesh is fine (y'^ < 1), the viscous sublayer is properly resolved (Wolfstein 1969). This hybrid wall treatment technique is for-mulated to produce reasonable answers, although the wall cell centroid falls w i t h i n the b u f f e r region of the boundary layer (5 < < 30).
This system of equations is solved using a segregated iteraUve solution method based on the SlMPLE-algoiithm (for details on discretization and solution methods, see Ferziger and Peric [2003]; Demirdzic and Muzaferija [1995]; Weiss et al. [1999]).
The numerical method and solvers employed for modeling the self-propulsion are the same as the ones used for the simulation of the bai'e hull flow. The flow field around the self-propelled hull is modeled at the same Froude numbers as the bare hull to facilitate comparison. The hull sinkage and trim are fixed to the measured ones f r o m the towing tank self-propulsion
measure-ments. To model the propulsion system of the wateijet geometry, an axial body force is uniformly distributed inside the volume containing the actual geometiy of the impeller. In this set of simu-lations, the rotor and stator are excluded from the pump geometry but the shaft and hub are still present. Excluding the hub from the simulations will cause flow separation inside the diffuser pait of the pump casing. The magnitude of the horizontal component of the body force is set equal to the computed total resistance of the entire system of the hull and the waterjet unit minus the rope force apphed to the hull during the self-propulsion test for unloading the propulsor due to the higher frictional resistance at model scale.
3.2. Hull and pump geometry
The hull used in this study is a planing hull designed by SSPA, with the model number 3209-B (Brown 2013). In this paper, this hull is refeiTed to as the SSPA hull. A body plan is presented in Fig. 2. The aft and fore perpendiculars are the sections marked by the numbers 0 and 20, respectively. Draft is measured at these sections. The towing point of the hull is 95 mm below the deck level and is located at Section 6. Particulars of the huU are given in Table 1.
For the self-propulsion test, the hull was equipped with a single wateijet propulsion unit designed by Rolls-Royce A B in Kristinehamn, Sweden. The nozzle exit diameter of the waterjet unit was 56.4 mm. The positioning of the waterjet unit on the SSPA hull is shown in Fig. 3. During the measurements, which were carried out i n the SSPA towing tank, the waterjet unit was mounted on the hull throughout both the resistance and self-propulsion test, but during the resistance test, the intake opening was covered and the unit was filled with water to create the same initial conditions for the bare hull and the self-propelled one.
3.3. Measurement technique
The method employed at SSPA to measure the flow rate through the wateijet unit is explained in this section. A T-junction, which should rednect the nozzle discharged flow to a direction peipen-dicular to the nozzle exit, is used to measure the momentum flux
//III
/ y
#/a I / //i / / / / / ^ ' 1 / / / / y yX^$}\////^^^'^
^
^
^
^
Fig. 2 Body plan of the hull used in this study
Table 1 Particulars of the hull
Scale factor a (—) 7.50
Length Lpp (m) 17.00 Beam B (m) 4.65
Length L W L (m) 16.98 Displacement (m^) 35 Draft forward Tp (m) 0.98 Draft aft T ^ (m) 0.98
Fig. 3 Positioning o f ' t l i e waterjet unit
Fig. 4 Flow rate measurement using T-junction
of the discharged jet. Through the axial momentum flux balance on the T-junction, the force exerted on this device is correlated to the jet f l o w rate. The T-junction is shown in Fig. 4.
Self-propulsion tests are caixied out in two stages. First, the waterjet-driven hull is run with free sinkage and trim and the pump shaft revolution is set such that the waterjet thrust balances the hull resistance (minus the rope force that is used f o r unload-ing the waterjet unit due to difference i n the frictional resis-tance of the model- and full-scale hulls). A f t e r this step, the T-junction is mounted on the h u l l and the sinkage/trim and the pump revolution are set to the values obtained f r o m the pre-vious step. The attitude of the hull is thus f i x e d by vertical bars, can-ying no load i n the x-direction. The force exerted on the T-junction is measured and used f o r obtaining the f l o w rate through the nozzle exit. A comparison of the pump shaft torque w i t h and without the T-junction shows very small differences and, therefore, running the hull at the same equilibrium position and the same operating condition for the pump as i n the self-propulsion measurements ensures that the f l o w rate with and without the T-junction remains the same.
In practice, there are some limitations and complications mea-suring the f l o w rate employing the T-junction method. Fkst of all, the T-junction needs to be up in the air to measure the jet momentum flux. In the lower speed range, the device is f u l l y submerged i n the water behind the transom or it may partially penetrate the transom wave. I n such cases, the force measured f r o m the T-junction does not represent the jet momentum flux. Besides, although the T-junction was built to redirect the f l o w at right angles to the nozzle discharged f l o w , during the mea-surements, it was noticed that the redirected f l o w is not exactly perpendicular to the discharged f l o w ; it bounced back shghtly toward the nozzle after hitting the T-junction. This means that the force exerted on the T-junction is larger than f o r the ideal case. I n order to correct this, the measured force on the T-junction was calibrated through bollard pull tests at various f l o w rates.
In Section 4.2, it w i l l be shown that the pressure at the nozzle exit is not atmospheiic. This pressure force component needs to be considered i n the T-junction momentum flux balance equation. Since the pressure distribution at the nozzle exit was not measured in the experiments, the pressure computed from computational fluid dynamics is used in the momentum flux balance equation.
3.4. Mesh generation and solution verification
In this section, the mesh used for the self-propulsion simula-tions is discussed and the computational setup verified. The cor-responding bare hull verification is shown in Eslamdoost et al. (2015). Like in the bare hull simulation, the flow around the self-propelled hull is considered to be symmetric with respect to the AZ-plane and therefore only half of the geometry is used in the simulation; this relies on an assumption that the jet is swirl free, which in normal operating conditions can be considered reason-able as is shown i n Section 4.2.1. The computational domain is extended l.SLpp in front of the forward perpendicular, 5.5Lpp behind the transom section, and 3Lpp i n the transverse direction. In the vertical direction, the domain is extended 2.5Lpp below and 1.5Lpp above the undisturbed free-surface level. For the sake of the mesh consistency between the bare hull simulations and the ones for the waterjet-driven hull, it is tried to keep the grid-generation technique as well as the mesh size and mesh distribu-tion are thé same as for the bare hull. However, the geometries are not exactly the same and a larger number of cells is required for the self-propulsion computational domain to generate the grid inside the waterjet unit. Besides, since the flow details into the waterjet intake and the discharged flow out of the nozzle are of interest, the mesh in these regions is refined further (Fig. 5).
Like for the bare hull, to obtain a suitable cell size and hence total grid number, a systematic grid refinement study was carried out for the self-propulsion simulations. Multiple grids with sys-tematically varied grid parameters were used at Froude number 0.798. The mesh convergence study was carried out by monitor-ing the resistance coefficient defined as follows:
C T = ^ ^ ^ , (15)
where Rt is the total resistance of the self-propelled hull (includ-ing the waterjet unit), p is the water density, S,e[ represents the wetted suiface of the hull at rest, and U is the ship velocity.
The total resistance coefficient convergence curve for the self-propelled hull is plotted in Fig. 6. Since the grid is unstructured, the step size hj on the horizontal axis is obtained as the third root of the total number of cells for grid i . The fmest grid is denoted by
III. The total number of cells for this grid is 16.2E6 and for the coarsest grid is 1.2E6. The times requked to run the simulations
Fig. 5 Grid distribution on ttie hull (left) and inside the ducting channel (right)
22ïl= 1 1 I I I 0 0.,1 I 1.5 2 2..Ï
Fig. 6 Convergence of the total resistance coefficient with grid refine-ment at Fn = 0.798 for the waterjet-driven hull. The full symbol shows the grid selected to perform the calculation for the entire range of Froude numbers. The p value is the power of the best f i t expression
f o r t h e error according to Ega and Hoekstra (2014)
on 64 processors (Intel, Xeon E5520, 2.27 GHz) are about 1.5 and 23 hours for the coarsest and the finest grids, respectively.
The formal verification was carried out based on the least squares root method by Ega and Hoekstra (2014). I n this method the iterative errors must be smaller than 1% of the grid eiTor. A good compromise between numerical accuracy and computational effoit was obtained with the grid shown by the full symbols i n Fig. 6, which was selected for further self-propulsion computations. The total number of cells for this grid is 5.7£'6 and the numeri-cal uncertainty f o r this grid is 2.3% of the computed resistance coefficient. The average y'^ values on the wetted surfaces for each refinement ratio are given in Fig. 7. The y'^ value f o r the finest and the coarsest meshes are around 10 and 130, respectively. The y'^ contour on the waterjet-driven hull at different speeds is shown in Fig. 8. The y'^ distribution on the wateijet-propelled hull is almost the same as on the bare hull at the same speeds (Eslamdoost 2014). This confurns the uniformity of the bare hull and self-propelled hull grids.
3.5. Self-propulsion computations
The simulated jet f l o w and the free-surface close to the transom are shown in Fig. 9. In Fig. 10, the axial body force distribution per unit volume as well as the pressure distribution on the symmetry-plane cut of the ducting channel are presented. The f l o w head
0 0.5 I l_5 2 2.5
/',/''!
Fig. 7 Average y + values for different self-propelled hull mesh refine-ment ratios at Fn = 0.8
increases as it reaches the region where the body force is applied. This region starts just before the expansion of the impeller casing and continues until the impeller casing reaches its maximum diam-eter. Due to the increased flow head, the pressure inside impeller casing and the nozzle increases but eventually adapts itself to the atmospheric pressure at the nozzle exit.
In order to calculate the gross thrust of the waterjet unit, the ingested momentum flux at the capture area and the momentum flux of the discharged jet at the nozzle exit are required. Since the flow field around the hull is resolved, it is possible to obtain the capture area geometry.
3.5.1. Computation of stieamtube and capture area. Using the same technique as Bulten (2006) and Ding and Wang (2010), the streamtube geometry is obtained by solving a convection equa-tion for a passive scalar in the reversed resolved velocity field. The boundary value of the scalar is set to 1 on the nozzle exit sec-tion and 0 on the other boundaries. Then, the scalar is obtained iteratively for the entire flow field. The cells inside the streamtube acquire the scalar value o f 1 and those outside the streamtube acquire 0. The scalar distribution on the symmetry plane is shown in Fig. 11. The transition of the scalar value f r o m 1 to 0 is not sharp and its spread depends on the grid size. As shown in Fig. 5, the grid close to the intake is refined to keep the transition zone as sharp as possible. The streamtube interface is extracted as an iso-surface of the scalar equal to 0.5 (Fig. 12).
The capture area geometry is obtained by extracting a vertical section along the computed streamtube which, as proposed by the
Wall Y+
,y I I M x i ').m r:.n j ^ i i
Fn = 0.2
IMJI III.O
W;ill Y+
mi) 4II.I) .W.O
F n = 0.8
Wall Y+ 12.(1 4a.{i
Fn = 1.4
Fig. 8 y + contour on the self-propelled hull at different speeds
Fig. 9 Simulated free-surface and jet flow at Fn = 0.797
FFTC Specialist Committee on Wateijets (ITTC 2005), is located one intake diameter ahead of the intake tangency point to the hull. The capture area geometry and the axial velocity contour at this section are plotted in Fig. 13 at different Froude numbers. I n order to validate the capture area computation method, the c o n -tinuity of the flow through the capture area and the nozzle exit needs to be satisfied. The ratio of the volumetric f l o w rate through the capture area to that of the nozzle exit is plotted in Fig. 14. As seen i n this plot, the deviation of the f l o w rate through the cap-ture area is less than ± 1 % of the computed f l o w rate through the nozzle exit. Through an analytical investigation, van Terwisga
40000
30000
2 i 20000
I
10000 -10000,O duct {upper part) < duct (lower part)
- ^
^
'A
O duct {upper part) < duct (lower part)
- ^
^
'A m / m / ; 1 i 1 i i - < 0.1 0.2 0.3 x [ r a ] 0.4 0.6Fig. 10 Pressure distribution on the symmetry-plane cut of the duct-ing channel at Fn = 0.797. The dashed line shows the axial body force
distribution per unit volume schematically
Fig. 11 Passive scalar contour at Fn = 0.797
Fig. 12 B o t t o m view of the hull showing the streamtube separating the flow into the waterjet intake f r o m the rest of the flow at Fn = 0.797
(1996) shows that an error of 1% in f l o w rate causes an error o f 2.4-3.5% i n gross thrust. A n even better match between the f l o w rate through the capture area and the nozzle exit can be achieved by refining the grid close to the intake, but this was not deemed necessary.
The influence of the waterjet-induced f l o w on the velocity field close to the intake was investigated by compaiing the velocity field in front of the intake for the waterjet-driven hull and the bare hull. I n order to make this shidy independent of the sinkage and trim differences between the bare hull and the self-propelled hull, the simulations were carried out at constant sinkage and trim values. The boundai-y layer profile at the symmetry plane of the computa-tional domain at the capture area as well as two other sections, one before and one after the capture area, is plotted in Fig. 15. Closer to the intake opening, the discrepancy between the self-propelled hull boundary layer profile and that of the bare hull
Fn = 0.199 Fn = 0.398 Fn = 0.434 Fn = 0A9i Fn = 0.598 Fn = 0.797 Fn = 0.995 0.00 0.210 Fn = 1.194 0.420 0.630 0.S40 Fn = 1.394 l.OS
Fig. 13 Nondimensional axial velocity (ui/uo) distribution on ttie capture area at different speeds
l-H 1.05 1.025 1 0.975 0.95. > € 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Froude Number
Fig. 14 Ratio of the volumetric flow rate through the capture area to the volumetric flow rate through the nozzle exit
|/W. , =2.4 intake ^/W. , =1 intake ^/W. , =0.40 intake 0.04 — bar ---scl e h u l l - p m p u l s i u n 0 0..Ï , I
Fig. 15 Boundary layer profile at Fn = 0.434 for three ^/lA/mtake ratios, where ^ is the axial distance of the section f r o m the intake tangency point and Wi„take is the intake width. ^ shows the normal distance from
the wall
becomes larger. The influence o f the intake suction is practically negligible 2.5 intake diameters in front of the intake tangency point to the hull.
The variation of the capture area width, i i ' i , measured f r o m the outer corners, at different Froude numbers is shown in Fig. 16. The width is almost constant at the lower and the intermediate Froude numbers but decreases at higher speeds. The ITTC Spe-cialist Committee on Waterjets suggests a constant capture area width f o r all the operating conditions (1.32 for a rectangular cap-ture area and 1.5 for a half-elliptical capcap-ture area [ITTC 2005]) but the cuiTent results are not supporting the constant capture area hypothesis. The difference between the mean axial velocity at the computed capture area and the one obtained f r o m the rectangular capture area is almost 2% at the design speed of the studied hull
(Fn = 1). The mean axial velocity of the rectangular capture area
is calculated using the computed velocity profile at the center of the capture area. According to the simplified gross thmst error
1.95r 1.9 1.9 e so 1.85
fi
1.85 1.8 1.75 1.7 l.ósj; 0 1 ( > e 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Froude Number
Fig. 16 Nondimensional capture area width over a range of Froude numbers
0.2 0.4 0.6 O
Froude Number
Fig. 17 Variation of tlie capture area to tlie nozzle exit area ratio over a range of Froude numbers
estimation method presented in Eslamdoost (2014), 2% eiTor in the average velocity at the capture area causes a similar error per-centage in gross thmst. Although the enor i n gross thmst estima-tion is 2%, the resultant error in thmst deducestima-tion fracestima-tion becomes 20%. This shows the large sensitivity of t on the mean velocity at the capture area especially in Froude number range where the thrust deduction fraction is close to zero.
The ratio of the capture area to the nozzle exit area is plotted in Fig. 17. The size of the capture area increases with speed up to Froude number of 0.43. Thereafter, it starts to decrease. This Froude number is i n fact the critical Froude number where the transom o f the waterjet-driven hull clears water.
3.5.2. Self-propulsion validation. There are several methods, which have been discussed in the literature for the measurement of the waterjet f l o w rate during self-propulsion tests (e.g.. Differ-ential Pressure Transducer, collecting the discharged water, and Laser Doppler Velocimetry). The measured f l o w rate and gross thmst presented in this paper are obtained by employing a classic f l u i d mechanics approach, called the T-junction method i n this paper. I f a free jet stream hits an obstacle and is then deflected such that the redirected flow direction is perpendicular to the origi-nal free jet stteam, the force exeited on the obstacle w i l l be equal to the momentum flux of the incoming jet stream. This force is measured in the experiments and then using some assumptions, the flow rate and the gross thrust of the wateijet system can be calcu-lated (Eslamdoost 2014).
The intake velocity ratio (FVR) and the nozzle velocity ratio (NVR), which are the mean velocity at the capture area and nozzle exit, respectively, divided by the ship speed, are shown in Fig. 18. The experimental N V R values, which are represented by triangular symbols, are obtained based on the measured flow rate through the nozzle, but the I V R values, which are shown by circular
sym-D^2.5 0^1.5 ^ 1 0.5, - n M m : C F D
VR:
C F D^R
:
E F DVR:
E F D i Von
m : C F DVR:
C F D^R
:
E F DVR:
E F D m : C F DVR:
C F D^R
:
E F DVR:
E F D ) < > •< 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 F r o u d e N u m b e rFig. 18 Comparison of measured and computed IVR and NVR
140 120 _ 1 0 0 ^ 80 - ^ 6 0 40 20 0, 1 ^ C F D • E F D < 4 1 ^ -< 0 0.2 0.4 0.6 0.8 1 1.2
Froude Number
1.4 1.6Fig. 19 Comparison of computed and measured gross thrust
bols, are obtained based on the empirical formula for the turbulent boundary layer on a flat plate and assuming the capture area to be rectangular (FFTC 2005). On the average, there is a discrepancy o f 2% and 2.7% between the computed and measured/estunated N V R and I V R , respectively. The FVR is almost constant over the entire studied Froude number range but N V R is very dependent o n the operating speed. Variations of N V R with Froude number have the same trend as the caphire area size (Fig. 17) and the humps in both curves occur at the self-propulsion transom clearance speed.
The computed and the measured gross thrust are plotted i n Fig. 19. Although the match between the computed I V R and N V R and those employed f o r obtaining the measured gross thmst is very good; the computed gross thmst is lower than the measured one. This highlights the importance of the accuracy of the flow rate computadon. Eslamdoost et al. (2014) show that a mere 1% underprediction o f the j e t velocity results i n 4 % undeiprediction o f the gross thmst.
3.5.3. Thrust deduction fraction computation. The thmst deduc-tion fracdeduc-tion definideduc-tion at f u l l scale was presented earher in equa-tion (6). The definiequa-tion needs to be modified at model scale to the following form:
1 (16)
where TJ^ii is the bare hull resistance and R^, is the rope force applied to the hull to unload the wateijet during the model tests to compensate for the difference between the frictional resistance at the model scale and ftill scale.
The measured and the computed baie hull resistance are plotted in Fig. 2 1 . It should be highlighted that the intake opening is closed on the bare hull. The solver and models used for the bare hull flow computations are the same as the ones used for the self-propelled hull computations described in Section 3.1. Further infor-mation about the bare huU flow computation and verification can be found i n Eslamdoost (2014) and Eslamdoost et al. (2015).
B y employing the gross thmst and bare hull resistance which are shown in Figs. 19 and 20, the computed and measured thrust deduction fractions may be obtained. These are plotted in Fig. 21. It is seen that the computed values are lower than the measured ones in the speed range where the measurements are carried out. The T-junction measurement method is a new and untested tech-nique and the results are strongly dependent on conection f o r nonorthogonaUty of the deflected flow and nozzle exit pressure. Preliminary computations of the T-junction flow indicated consid-erably larger corrections f o r nonorthogonaUty. This would bring
160 120 RO 40 -e-C • E F D F D 1
/
1A
0.2 0.4 0.6 0.8 I Froude N u m b e r 1.4 1.6Fig. 20 Comparison of tiie computed and the measured bare hull resistance 0.3 0.2 0.1 -0.1 -0.2 : C F D • •^ • T? T m _ ' e 0 0.2 0.4 0.6 0.8 1 1.2
Froude Number
1.4 1.6(equations [9] and [10]) are computed and shown at different speeds. Finally, the relative importance of the intake and the exit drags is discussed.
4.1. Intake drag
For the investigation of the intake drag, it was assumed that the shear stress acting on Ai, A2, and A4 is negligible and only the pressure resistance on these surfaces was taken into account. The pressure force was split into hydrostatic and the hydrodynamic components to track the source of the drag more in detail. These pressure components at Fn = 0.8 are shown in Figs. 23 and 24, respectively. Contiibutions of the hydrostatic and hydrodynamic pres-sure forces to Di fi'om A^, A2, and A4 at different Froude numbers are shown in Fig. 25. Note that a positive contribution is obtained for a force in the negative .v-direction for A, and A2, but in the positive direction for A4. The hydrostatic pressure force com-ponent is larger than the hydrodynamic force exerted on these surfaces. Since the projected area of / I , and A2 on the ^z-plane is much larger than that of A4, these two surfaces have the largest hydrostatic pressure force components; but since the nonnal vec-tors to these surfaces have their longitudinal components in oppo-site directions, these tvvo large hydrostatic force components have different signs and cancel each other to a lai-ge extent. The hydro-static pressure force on Ai and A2 is largest in the lower Froude Fig. 21 Comparison of the measured and the computed thrust
deduc-tion fracdeduc-tion
the measured data closer to the computed ones. Further investiga-tions of the T-junction technique are planned. It should be pointed out that accurate measurements of the thrust deduction are very difficult (Eslamdoost et al. 2014), where the measured thnist deduc-tions for the Athena hull in a comparative ITTC campaign varies between -|-0.08 and -F0.16.
4. Computation of intake and exit drag
The method for the computation of the capture area and the streamtabe was already discussed in Section 3.5.1. In the follow-ing sections, usfollow-ing the surfaces of the capture area, streamtube,
A4 (Fig. 22), and the nozzle exit, the intake and the exit drag
Fig. 24 Hydrodynamic pressure coefficient on the capture area, the streamtube and A4 at Froude number 0.8 (bottom view)
-1 -a.;^-.--B|Tt:--4-.--. B- 9 : : : r : : : : : : | r . : -B _---> -1 0 - A | : Hydrodynamic -Q-Ay Hj dfostalii: -J>A^: Hydrudynamic - ^ A ^ : Hydrostaiic - • - A j : HydiodynLimic -B-A : Hydrostalrc •I ! 1 1 I I • I " " " I I ü 0.2 0.4 0.6 O.S 1 1.2 1.4 1.6 Fi'oude N u m b e r
Fig. 25 Hydrostatic and hydrodynamic pressure contributions to the intake drag
number range because of the deeper submergence of the hull in this speed range (Eslamdoost 2014). There is also a larger capUire area and projection of the sfreamtube on the j'z-plane. The varia-tion o f the caphire area with speed was shown eariier in Fig. 16.
4.2. Exit drag
The exit drag is caused by nonatmospheric pressure on the jet exit. In the literature (van Terwisga 1996; I T T C 2005), this com-ponent is assumed significant only i f the exit is submerged. How-ever, even i f the jet is ejected into the air, the pressure w i l l be nonatmospheric, unless the streamlines leaving the nozzle are exactly parallel. This is unlikely to be the case for most nozzles, and for the SSPA hull, a relatively large pressure is found in the jet center. The pressure dishibuhon is shown i n Fig. 26. Integrating this pressure over the nozzle exit area, the exit drag is obtained.
4.2.1. Swirl effect o n e.\it drag. The f l o w shown in Fig. 26 is
swirl free. Tangential forces are neglected i n the present body force model. This assumption can be violated in off-design condi-tions where the stator is not functioning in its optimum condition. In such cases, the pressure at the nozzle exit can be different from the case that has the same f l o w rate but no swirl. The effect o f swiri is discussed by the ITTC Specialist Committee on Wateijets (FFTC 1996), but there is no information given about swiri levels in existing wateijet systems. There is, in fact, very little mforma-tion about swirl available in the hterature. Two reports which do include such information are, however, those o f Rispin (2005) and Jonsson (1996). The tangential component o f the nozzle exit velocity is very dependent on the operating condition o f the pump. Rispin (2005) shows that at the lower flow rates, where the pump
^ P n a a u r a ( P a )
.Y ^ -l.QgHa 2.0*+02 1.Je*ö3 2.6*+03 ,1 St^Hja 5.0^.03
Fig. 26 Total pressure distribution at the nozzle exit section at Fn = 0.5
and the stator are at off-design conditions, some swirl is left in the nozzle exit flow, while close to the design flow rate, the tangential velocity of the jet is neariy removed, which indicates that the stator performs its task veiy well.
According to the measurements presented by Rispin (2005), the average tangential velocity over the nozzle exit section varies between 4% and 17% o f the average axial velocity at different operating conditions. Observing the angle of some strings placed at different radii of the nozzle exit at different pump operating conditions, Jonsson (1996) indicates that the ratio o f the tangen-tial velocity to the axial velocity at the nozzle exit is at most 17%, where measurements were possible. The swirl is maximum at the jet center due to a center vortex created by the hub, but neither in Rispin (2005) nor i n Jonsson (1996) the swirl at the center is reported. The reasons are that i n the study carried out by Rispin, the hub was extended to accommodate the water-ttannel rear drive shaft arrangement, so there was no center vortex started at the nozzle exit, and in the investigation by Jonsson, the rotation was so high at the center that it was impossible to measure any reliable angle for the visualizing string. But since the area close to the center is so small, the strong swirl i n this region does not sig-nificantly affect the area-weighted average of the swiri over the entire nozzle exit area.
A n evaluation of the swirl effect on the pressure force acting on the nozzle exit was accomplished by adding tangential com-ponents to the body force applied to the impeller region in tiie numerical simulation of the f l o w around the waterjet-propelled hull. Since after the introduction o f the tangential component o f the body force, the f l o w was no more symmetric, both sides o f the hull had to be included. It was assumed that the tangential component of the body force is equally distributed in the radial direction inside the volume containing the impeller. As the rela-tion between the tangential component o f the body force and the jet swirl at the nozzle exit is not known, f r o m the beginning it was required to try out some different tangential body force mag-nitudes in order to obtain the desired swirl at the nozzle exit. On the basis o f the measured swirl angle in Rispin (2005) and Jonsson (1996), the tangential component of the body force was specified so as to obtain a 10° of area-weighted average swirl at the nozzle exit. According to the mentioned references, 10° swirl (tangential velocity 17% o f axial velocity) is the maximum mea-sured for the studied waterjet pumps and, therefore, the pressure difference obtained based on such a swnling flow should be con-servative. When the waterjet pump operates around its optimum condition, its efficiency as well as the swirl at the nozzle exit are almost independent o f the craft speed and remain unchanged. Therefore, even at off-design conditions, 10° of jet mean swirl should still be a conservative assumption.
The computed swirl vector indicating the magnitude and the direction o f the velocity vector component on the ;'z-plane is shown i n Fig. 27. The area-weighted swirl average is 10°. The pressure disttibution at the nozzle exit is shown i n Fig. 28, which should be compared with Fig. 26. Since the swirl is strongest at the jet center, the largest pressure reduction can be noticed in this region. The inh'oduced tangential body force inside the impeUer chamber increases the flow head comparing to the case which has only an axial body force component. Besides, the pressure dis-tribution on the nozzle exit influences the f l o w rate through the nozzle. For the computed case, the f l o w rate o f the jet with swirl is 0.5% larger than that o f the swirl-free jet. Integrating the
Swirl
0.0 OM 0.83 1.3 1.8
Fig. 27 The swirl vector showing the magnitude of the radial and the circumferential velocity components at the nozzle exit
P r e s s u r e ( P a )
.I.Oc+03 :^.0c+01i l . l e + 0 3 2.6C403 3.Bc+03 5.0e+0a
K 1 Fig. 28 The total pressure distribution at the nozzle exit section for the
jet with 10° of inclination with respect to the axial direction due to swirl
pressure distribution on the nozzle exit f o r the swirl-free and swirling jets reveals that the swhl results in 19% smaller pressure force acting on the nozzle exit area. It should be noted that such a reduction of the exit drag may occur only in extreme off-design conditions. A t design conditions, it should be considerably smaller.
4.3. Relative importance of intake and exit drags
Figure 29 shows the intake and exit drags versus Froude number. Except at low speed, the intake drag is positive, while the exit drag is negative in the entire speed range. In the major pait of the speed range, the exit drag magnitude is larger than the magnitude of the intake drag, so the largest contribution to the d i f -ference between the gross and net thaists is the exit drag. The exit drag magnitude increases more or less linearly with speed, but the intake drag magnitude increases first and then starts to decrease shghtly in the Froude number range 0.6 < Fn < 1.4. The
magni-Fvoudc Niiruber Fig. 29 Intake and exit drags
Fi-oiiilo N i i i i i l i o r
Fig. 30 Comparison of the difference between the gross thrust and the net thrust with the sum of the intake and the exit drag
0
j l , : . . , , . 1
0 0.2 0.4 0.6 0.8 I 1.2 1.4 1.6 F i o u d c N i i i u l j c r
Fig. 31 Jet thrust deduction fraction
tude of the sum of the two components has a peak at the transom clearance Froude number but increases monotonously (in the nega-tive direction) above Fn = 0.5.
4.4. Jet thrust deduction fraction
According to equation (8), the sum of the intake and exit drags is equal to the difference between the gross and net thrusts. I n practice, there w i l l be a difference due to the limited numerical accuracy. The thrust difference is computed f r o m two large numbers: the gross thrust obtained f r o m the momentum balance and the resistance of the hull in self-propulsion. These numbers are of the same order of magnihide, while the difference is one order smaller. The intake and exit drags are obtained from forces acting on small areas of the total system. In Fig. 30, the difference between the thrusts is compared with the sum of the intake and exit drags. Considering the numerical accuracy, the correspon-dence is very good.
As shown in equations ( I I ) and (12), the jet thrust deduction fraction, tj, can be obtained f r o m the difference between the gross and net thmsts (implicit formula) or f r o m the sum of the intake and exit drags (explicit formula). Figure 31 shows the jet system thrust deduction fraction computed by both methods. According to this figure, the magnihide of tj is largest at very low speeds but decreases very fast as the Froude number increases. The smallest magnitude of tj occurs around Froude number 0.5, which is the transom clearance Froude number of the bare hull. After this speed, the tj magnitude increases slightly.
5. Conclusions
The objective of this paper is to explain the relation between net thrust and gross thrust of a wateijet system. For this purpose.
a technique using a RANS method with a V O F free-suiface rep-resentation combined with a body force reprep-resentation o f the pump has been developed. Validation showed good agreement for f l o w rate, but the gross thrust and thrust deduction were smaller than those f r o m the measurements. A possible reason for the discrepancy is the unconventional experimental technique with a T-junction to measure the flow rate.
The difference between the gross and the net thrust o f the studied wateijet system has been computed f r o m two different approaches. The first approach is a direct calculation o f the dif-ference through subtracting the net thrust f r o m the gross thrust and the second approach is to compute the intake and exit drags. Theoretically, the sum o f the intake and exit drags is equal to the difference between gross thrust and net thrust. The main dis-tinction between these two approaches is the magnitude o f the forces involved. I n the first approach, the difference is obtained by subtracting two large numbers, while in the second approach small numbers are summed up. I n spite o f this, the results are very similar over the studied Froude number range.
The magnitude o f the difference between net thrust and gross thrast increases as the speed increases, but when the difference is expressed in a nondimensional form, the jet thrust deduction frac-tion, it is seen that the difference is largest i n lower speed range but becomes smaller i n the intermediate- and high-speed range. The results f r o m the second approach show that the largest con-tribution to the difference between the gross thrust and the net thrust comes f r o m the exit drag due to the nonatmospheric pres-sure at the nozzle exit, which depends on the nozzle shape and whether or not the exit is out of the transom wave.
During the investigations cairied out in this study, several addi-tional results were obtained. The geometry of the caphire area and the streamhibe were computed as well as the exit pressure and the vena contracta. It was found that the caphire area is quite different f r o m the rectangular or semielliptical shape suggested by the ITTC Specialist Committee on Wateijets (FTTC 2002). Despite this, the average velocity o f the rectangular capture area obtained f r o m the boundary layer profile on a flat plate is very close to the com-puted average velocity on the actual capture area. The area and the width of the caphire area vary with the speed. The area is largest at the critical Froude number where also the resistance coefficient has a maximum.
The diameter o f the vena contracta is 3% smaller than the nozzle exit diameter and i t is located less than one nozzle exit diameter after the exit section. The diameter and the location o f the vena conttacta are dependent on the nozzle geomehy.
According to the computed momentum flux correction coeffi-cients, the assumption o f a uniform velocity distribution w i l l have the largest influence on the momentum flux at the capture area. This assumprion w i l l result i n 1-2.5% underprediction o f the mgested momenUim flux. The momenUim flux at the nozzle exit will be 0.5% underpredicted. Neglection o f these correction coeffi-cient results in 0.1-0.9% error in gross thrust and thus 1.5-9% error in thrust deduction fraction.
Acknowledgment
The research presented has been sponsored by Rolls-Royce Marme through the University Technology Centre at Chalmers.
The measurement data presented were obtained from SSPA. Com-puting resources were provided by C3SE, Chalmers Centre for Computational Science and Engineering.
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