Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics (1994) / Chapter Skim
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Session 10- Lifting-Surface Flow: Inviscid Methods
Session 10- Lifting-Surface Flow: Inviscid Methods
Pages 509-558
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From page 511... ...
The kinematic and dynamic boundary conditions, which are fully three dimensional, nonlinear and time-dependent, are satisfied on an approximate surface consisting of the propeller surface beneath the cavity and the portion of the blade wake surface overlapped by the cavity. An efficient and robust algorithm is developed to predict arbitrary cavity shapes, including so-called "mixed" cavity planforms in which the blade is partially cavitating at inner radii and supercavitating near the tip.
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From page 512... ...
Also, the importance of the crossflow terms in the dynamic boundary condition may be investigated. Finally, the method should be interrogated to determine if it predicts multiple solutions.
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From page 513... ...
Figure 3: The approximate cavity surface on which the boundary conditions are applied. dipole distribution on the trailing wake surface, Sw(t)
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From page 514... ...
In the fully nonlinear solution, subsequent iterations are found by satisfying the dynamic boundary condition on an updated cavity surface, where the kinematic boundary condition is used to update the surface, as described in section 2.3. The solution is then considered to be converged when the cavity surface does not change (to within a tolerance)
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From page 515... ...
2.2 Dynamic Boundary Condition (8, The dynamic boundary condition (DBC) requires that the pressure everywhere inside and on the cav ity be constant and equal to the known cavity pres sure, Pc.
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From page 516... ...
However, this is not the case since the cavity curvilinear coordinates are approximated with those on the propeller surface. Nevertheless, in am plying the dynamic boundary condition, the normal velocity is assumed to be vanishingly small.
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From page 517... ...
. 2.3 Kinematic Boundary Condition Since the dynamic boundary condition is applied on the portion of the boundary which is encompassed by the cavity, the other boundary condition, namely the Cinematic condition, may be used to determine the position of the actual cavity surface once the singularity strengths have been determined.
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From page 518... ...
2.3.2 KBC on the Walce Cavity The kinematic boundary condition on the cavity surface in the wake may be derived in a similar fashion, except that now both surfaces of the supercavity must be considered Do in-g+ is, a, t)
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From page 519... ...
, the dynamic boundary conditions (15)
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From page 520... ...
The blade and trailing wake discretization is identical to that in the case of fully wetted- unsteady flow [8, 17, 16~. If we call NWS = Number of wetted blade panels NCB = Number of cavitating blade panels NCW = Number of cavitating wake panels, then, among the discrete sources and dipoles, we have NWS known source strengths, via (31)
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From page 521... ...
The system of inde~cing is shown in Figure 9. A single discrete equation may be written to replace the dynamic boundary conditions on the blade (15)
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From page 522... ...
The height of the cavity at its trailing edge, b(r, t) , will in general be non-zero, unless we have guessed the correct cavity planform.
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From page 523... ...
The cavity closes at all spanwise strips for only one solution, and this is the correct clarity planform for the given cavitation number and operating conditions. For arbitrary cavity planforms, one must address the possibility that the cavity trailing edge does not coincide with a panel boundary.
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From page 524... ...
While this does not necessarily preclude the existence of multiple solutions, it does show that, if they exist, they are difficult to find using this method. 4.2 The Crossflow Terms It was mentioned in section 2.2 that the crossflow terms ~ and Vu+ in the dynamic boundary conditions (15)
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From page 525... ...
Figure 17 shows a typical blade and hub discretization. For the cavity solution, the hub is assumed to be fully wetted and the following kinematic boundary condition is applied: (t)
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From page 526... ...
and those computed by linear theory and linear theory with leading edge corrections (labeled PUF-3A [11, 12~. The computations are done for steady flow conditions on the one-bladed AO-177 propeller.
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From page 527... ...
5 Conclusions A potential based boundary element method has been developed for the analysis of unsteady sheet cavitation for propellers of extreme geometry. The method is able to predict, in an efficient and robust manner, arbitrary unsteady cavity planforms on a blade discretization which is fixed in time.
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From page 528... ...
This could be accomplished by treating the tip vortex as an inner problem, the solution of which should be matched to the outer solution from the present method. The inner problem could be treated by a boundary element method with a grid which is chosen to fit a vortex with an assumed core radius (possibly determined semi-empirically)
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From page 529... ...
Hsin. A boundary element method for the analysis of the unsteady flow around extreme propeller geometries.
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From page 531... ...
For other cases however, especially for lifting surfaces with wide circular tips (similar to those of propeller blades) , the circulation distributions predicted by the BEM have been found not to be consistent to those predicted by the VLM A
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From page 532... ...
The conventional grids for a propeller blade and a circular planform hydrofoil are shown in Figures 1 and 2, respectively. Figure 1: The conventional grid on a propeller blade and its trailing wake.
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From page 533... ...
Notice the large difference over the span between the circulation distributions before and after the IPK condition, especially for the 20% thickness to chord ratio CPH, shown in Figure 4. This large difference in circulation has been caused by the relatively small adjustment of the trailing edge pressures in the vicinity of the tip, as shown in Figure 5.
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From page 534... ...
Cp = (p - poo) /P/U2/2 Figure 6: The blade orthogonal grid on a circular planform hydrofoil and its trailing wake.
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From page 535... ...
This explains the larger difference between flee circulation distributions before and after the IPK condition, in the case of the blade orthogonal grid than in the case of the conventional grid, stated earlier. In the case of the blade orthogonal grid the angle between Vm and the s direction is larger than in the case of the conventional grid.
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From page 536... ...
maX = 0.2, ~ = 5.73°. Predicted by applying the BEM on the blade orthogonal grid; after the IPK condition.
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From page 537... ...
The resulting total velocity flow field and the corresponding streamlines6 for the circular planform hydrofoil with 20% thickness to chord ratio, are shown in Figure 11, where the expected contraction in the wake shape can be clearly seen. The shape of each of the streamlines may be determined by integrating the velocity flow field shown in Figure 11.
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From page 538... ...
Notice that the circulation distributions before and after applying the IPK condition are closer to each other than they were in the case of the conventional and the blade orthogonal grid. Also notice that the circulation distribution now extends only up to the location of the computational tip and that its values after the IPK condition does not show the previously observed "peculiar" behavior at the tip.
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From page 539... ...
Predicted by applying the BEM on the flow adapted grid; before and after applying the IPK condition. 539 Figure 17: Velocity vectors on the suction and pressure sides at the trailing edge of the circular planform; tT/Cima~ = 0.2, cat = 5.73°.
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From page 540... ...
The method is summarized in the following steps: · Solve the zero thickness VLM problem and determine the strengths, L's, of the line vor tices, by satisfying the kinematic boundary condition at appropriately selected control points on the planform: Vr n =-Uon (16) where Vr is the velocity vector induced by all discrete vortex horseshoes on the plan form and its wake; Con is the component of the inflow normal to the planform, i.e.
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From page 541... ...
Notice the large difference between the circulation distributions in the case of the conventional and the blade orthogonal grid. This is a consequence of the fact that the effect of velocities due to thickness on the Kutta condition, is ignored in the VLM.
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From page 542... ...
~ \ 0.0 ....,....,....,... o.oo 0.25 0.50 0.75 Y/Y~x 1.00 Figure 19: Circulation distributions predicted from BEM (after the IPK condition)
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From page 543... ...
When a flow adapted grid was applied on a circular planform hydrofoil, the performance of the boundary element method was found to improve substantially. In addition, the results from applying a vortex-lattice method on the same grid, were found to be in very good agreement to those from the boundary element method.
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From page 544... ...
Hsin. A boundary element method for the analysis of the unsteady flow around extreme propeller geometries.
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From page 545... ...
A nonlinear unsteady pressure-type Kutta condition is applied that enforces zero blade trailing-edge loading at each time step. Example problems of the nearencounter between a vortex and an airfoil, as well as unsteady loading on a foil by upstream flapper foils demonstrate the validity of the techniques.
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From page 546... ...
where ub is the velocity of the blade surface. This boundary condition is augmented by the Kutta condition, which requires zero pressure loading and permits a tangential slip velocity at the trailing edge.
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From page 547... ...
The governing equations then are transformed into a velocity potential form and are enforced at panel control points located at the center of each panel. The boundary condition is implemented by directly substituting un from Eq.
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From page 548... ...
= 0 . Since the impermeability boundary condition is applied on A, the flow on the upper and lower surfaces is tangent (superscript s)
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From page 549... ...
(21) 1/2puo The sub-iteration formula at each time step to update shed vorticity strength is then given by Yw Y W MY W Vortex Core Splitting and Merging (22)
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From page 550... ...
(7~. The capability for accurately modelling shed vorticity was demonstrated by comparing the predicted wake behind an oscillating airfoil with experimental measurements (15~.
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From page 551... ...
Figure 1 compares the trajectories of the free vortex at representative time steps for both methods. Because Chow's analytical solution extends only a chord length downstream of the airfoil, the comparison is limited to this region.
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From page 552... ...
3. The box measurements were intended to provide upstream and downstream boundary conditions for Navier-Stokes computations.
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From page 553... ...
The foil transient lasts about one period and the calculation extends for six periods with 50 time steps in each period. The corresponding time-averaged mean pressure distributions from the unsteady calculation (reduced frequency of 3.62)
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From page 554... ...
-0.4 -0.5 ~.s `, -0.6 SUCnON SIDE _ ~EXP, xh-0.S88 CALC, ~c-0~391 ~CALC (HIGH 0, ~c=~S95 c3-0.6 -0.7 EXP, xh-0.S88 CALC.
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From page 555... ...
Comparisons of amplitude and phase angle of the first harmonic of C p TIME= 1 0 O O {:C; TIME=300 / TIME= 1 00 Figure 10. Snapshots of predicted vortex patterns at each time step for UTRC's stator-rotor turbine -2 _ -6 _ -8 -r -10 .12 1 _ _ _ -14 _ ~ -6 _e 1 n .
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From page 556... ...
Wake vortex shedding was neglected. The averaged pressure distributions from both the quasi-unsteady calculation and the fully unsteady calculations are in good agreement with experimental measurements.
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From page 557... ...
8. Chow, C.Y., and Huang, M.K., "Unsteady Flows About a Joukowski Airfoil in the Presence of Moving Vortices," AIAA Paper No.
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From page 558... ...
For a typical calculation with a reduced frequency of 10, a normalized unity chord length and freestream velocity, and 25 time steps In one calculation period, cods is about 80. On the average, we use 25 to 50 time steps and 160 to 300 panels to represent periodic foil motion in unsteady flows.
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