Twenty-Third Symposium on Naval Hydrodynamics (2001) / Chapter Skim
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An Unsteady Three-Dimensional Euler Solver Coupled with a Cavitating Propeller Analysis Method
An Unsteady Three-Dimensional Euler Solver Coupled with a Cavitating Propeller Analysis Method
Pages 616-638
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From page 616... ...
ABSTRACT A fully three-dimensional unsteady Euler solver, based on a finite volume scheme and the pressure correction method, is developed and applied to the prediction of the effective wake for propellers subject to non-axisymmetric inflows. The propeller is modeled via unsteady body force terms in the Euler equations.
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From page 617... ...
The propeller loading is needed to determine the body force distribution in the Euler solver, and the propeller induced velocity is needed when the effective wake is calculated. If unsteady sheet cavitation exists on the blade, then the extent and the thickness of the cavities are also determined as part of the solution.
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From page 618... ...
3.] Steacly Euler Solver In the steady Euler solver, named WAKEFF-3D, the method of artificial compressibility (Chorin 1967)
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From page 619... ...
On this surface, a periodic boundary condition is applied. 3.2 Unsteacly Euler Solver In the unsteady Euler solver, named WAKEFF-3U, the finite volume method is applied only for the dis
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From page 620... ...
_ ,u2 'led = v , F= ov , G= v2 , w ow vw ow -UP/ + fx H = vw , Q = —~P/0Y + fy w2 -0P/0Z + fz Note that the pressure gradient terms are grouped in Q with the body force terms because the pressure correction method is adopted in the unsteady formulation. From this point, the formulation and the numerical implementation of equation (17)
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From page 621... ...
From the dimensionless Euler equations for two-dimensional, incompressible, steady flows, the following expression for the body force can be derived for the cell shown in Figure 6.
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From page 622... ...
, so that they can exactly cancel the convective terms on the left hand side of the momentum equations, when the body forces are substituted into the ........... pi _ 035 _ 025 _ 015 ~~b ~ 055 Figure 9: Stream lines and pressure contours of the Euler solution (surface distribution model)
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From page 623... ...
In Figure 9, the solution of the Euler equations is shown, as obtained with the body force distribution shown in Figure 8. The formulation of the two-dimensional Euler solver is similar to that of the three-dimensional problem explained earlier, when the z coordinate is omitted.
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From page 624... ...
The pressure field and the streamlines obtained from the Euler solver are similar to those of the potential flow solution shown in Figure 10, even though the foil thickness effect is completely lost in the Euler solution. The difference of the flow fields computed by the potential solver and the Euler solver is shown in Figure 13.
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From page 625... ...
n 1 o of -n no · · -0.1 Figure 14: Comparison of the velocity magnitudes predicted by the Euler solver and the panel method along the dashed line shown in Figure 13; 50x50 cells, body force from the camberline pressure model, using equation (344. there is only local difference in the solution if this distance is varied from one cell to four cell thickness for the usual mesh of which the cell size near the disk is 0.04.
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From page 626... ...
The product of the pressure, the unit normal vector, and the area is calculated first for each side, and then the two force vectors from each side are added to obtain the force vector corresponding to the element on the camber surface. An actuator disk with thrust loading coefficient CT=3.0 iS analyzed using the unsteady Euler solver.
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From page 627... ...
, the flow field is more or less fully developed, and thus the changes begin to decrease to reach a certain level limited by the machine accuracy. In Figure 17, the time evolution of the axial velocity and the pressure is shown.
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From page 628... ...
This means that the flow field obtained by the unsteady solver is not consistent with the flow field predicted by the propeller potential flow solver simply because the propeller potential flow solver does include the blade thickness effect. Placing the body force cells on the "exact" blade surface, using the present discretization in the Euler solver, is practically impossible due to the fact that the resulting size of a cell is comparable to the blade thickness.
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From page 629... ...
, shows that the difference between the two flow fields, predicted by the steady Euler solver and the potential flow solver, is as small as 1% of the inflow velocity. This small error can be explained from the fact that the time averaged effect of the blade thickness is small as compared to the local instantaneous effect which can be larger.
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From page 630... ...
The zero normal velocity boundary condition is applied on the hub in the Euler solver, while the hub is not included in the potential flow solver. Even though the potential flow solver has the ability to consider the hub via a simplified image model, the hub is turned off in the present potential solver computation because the simplified image model has not been found to produce the correct flow field upstream of the propeller Total 0.5 Effective ~1 ~ T~1~L 11 no UK: 095 097 099101 103105 Am' z z - 1 z Figure 23: Comparison among the total, the propeller induced, and the effective wake as predicted by the unsteady Euler solver at 0.323 radius upstream of the zero-thickness One-bladed DTMB 4148 propeller.
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From page 631... ...
~ _ _ Ed ~ Al A 1~ ~ Zi "El ;9 _~ _ ~~ ~ ~ J~ ~ ~ ~ ~~ 0~o 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -200 -150 -100 -50 0 50 100 150 200 Angular Position (dog) Figure 24: Axial velocities predicted by the unsteady Euler solver and by the potential flow solver for the one-bladed DTMB 4148 propeller at ~eff=~0.323.
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From page 632... ...
10-' 10-0 10-6 10-7 ~ WAKEFF-3U 1 o-8 A MPUF-3A o Effective Wake 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -50 0 50 100 150 200 Angular Position (deg.) Figure 26: Axial velocities predicted by the unsteady Euler solver and by the potential flow solver for the zero-thickness DTMB 4842 propeller; ~eff=~0.323.
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From page 633... ...
0 8 0 4 n ~ 0 0.25 0.5 0.75 A3, Vx Figure 32: Comparison of the mean and the third harmonic of the time averaged unsteady effective wake. meets that the unsteady effective wake does not vary very much with time, even though the total and the propeller induced velocity do change.
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From page 634... ...
The cavitation can be predicted by the propeller potential flow solver once the effective wake is obtained through the iterations between the Euler solver and the potential flow solver. It should be noted that in the evaluation of the effective wake the propeller induced flow field has been determined by excluding the influence of the thickness and cavity source.
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From page 635... ...
6 CONCLUSIONS Robust steady and unsteady Euler equation solvers, coupled with a previously developed propeller potential flow solver, are developed for the prediction of the steady and unsteady effective wakes. To the author's knowledge, the present unsteady Euler solver is the first method that ever computes the unsteady eJective wake with sufficient space and time resolution.
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From page 636... ...
AND K~NNAs, S 1998a A 3-D Euler solver and its application on the analysis of cavitating propellers.
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From page 637... ...
Thus, in the steady flow case, one only needs to couple the liftingsurface code with an axisymmetric RANS or Euler solver. The authors' conclusion would seem to justify this assumption.
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From page 638... ...
, cm be obtained from the iteration betw en the steady nonaxisymmetric Euler solver Ed the lifting su face code Note that the body force used in this steady analysis is The time averaged propeller force, which is varying in pace here is yet mother prediction by using en axisymmetric Euler solver, m which The velocity flow field is c function of only the radius, q~(~) Inthe axisymmetric crurlysis, the body force is obtained by circumfe~enticlly averaging the propeller force One of The conclusions m the paper is that the time average of q~(r,t)
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