Twenty-Third Symposium on Naval Hydrodynamics (2001) / Chapter Skim
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Reynolds-Averaged Modeling of High-Froude-Number Free Surface Jets
Reynolds-Averaged Modeling of High-Froude-Number Free Surface Jets
Pages 848-862
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From page 848... ...
A''' Stokes equations and the code CFDSHIP IOWA, is described where the standard k e tub bulence model has been augmented with an nigh brnic P y old stress model to capture near surtacr stress nnisotropJ in addition, turbulence generated w wes have been described using n wa:v~nction model with n source term for the energy transfer from the sub su face turbulence to unsteady free surface dim turbancm The effect of the su face fluctuations on the flow are modeled -- an npprcximate shear stems boundary condition at the free surface Pmults are presented for predictions of the mean fl w and turbo lence quantities, as well as fre~surtace fluctuations with comparison to experimental data 1- Introduction Engineering predictions for turbulent flow near n free surface are of inter t in applications ranging from ship hJdrodJnamim to manufacturing processes The nonlinear nature of free su face boundary conditions along with the nonlinearity of the underlying Na:vier Stokes equations, make this problem anal tically in ton table, an d computationally challenging as w 11 A major impediment to the accurate modeling of tur bulent free surface flow, i e fl ws with n gas liquid inte face, has been the lack of an appropriate form for the P Jnoids weraged equations Unsteady surface elevation fluctuations cause the boundary to be in di tinct, or'fuzzJ, in the conte t of the weraged set of equations PecentiJ this issue was nddrersed in Hong & Walker (2000) where, starting from the gen ernl, variable property N
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From page 849... ...
The predicted surface fluctuation distribution is in good agreement with the observations Finally, a two w J coupled calculation is performed, where the surface fluctuations are used in the boundary condi tions on the subsurface fl w This agrees Ims well with the experiment, due mainly to z lack of an ef fective, rational method for switching between the low F oude number and high F oude number treat ment of the free su face 2 Governing Equations 2.1 Reynolds Averaged Navier Stokes Equations In this section, the Reynolds weraged N A -- - Stoker equations for application to the problem of turbulent free surface flow are presented They are then ape cialized to the case of small surface fluctuations, and an zpproaxi mate set of free surface bound ary condi tions are derived The Exact Averaged Equations The beEwtor of the mean fl w is described by the P y old veraged form of the N
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From page 850... ...
For the unkn wns beJond the PeJnoids stresses, i pproprti te turbulence models must be developed to i 11 w the set of equi tions to be closed Approximate Equationz for Small Surface Fluctuationz One would like to hi e i n i pprcximi te form of th x equi tions which cl n be solved using i conventioni. PANS solver of the tJpe empioJed in ship hJdrodJ ni mics To obti in this, the derivitiv i conti ining the vi rti ble mei n fluid prope ties in the momentum equi tions i re expi nded, i nd then the terms involv ing gri dients of the fluid properties i re collected to gether This Jioids i momentum equi tion of the form :SU\ ~ SU\1 SP _ 3u; PL e+7i5~i]
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From page 851... ...
/2 (21) 2.2 Surface Fluctuation Model Wav - Action Spectrum Model The unsteady free surface fluctuations are modeld up ing a w we action spectrum model like that described -- Komen Be cl (1994)
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From page 852... ...
~ k 7/3 (29) where k is the three dimensional w e vector, and k = k This three dimensional spectrum is inter 3 Numerical Implementation The computations for this study were carried out up ing n modified version of CFDSHIP IOWA Changes were made to incorporate the nen~surtace stress nnisotropJ, coupled solution of the w we n tion spec trum model, and the npprcximate dynamic free surface boundary condition which relates the Spar ent stress on the su face to gradient in the surface fluctuation variance Modeling of the near surface stress nnisotropJ was accompli Led in the context of the basic k e model which is included in CFDSHIP IOWA The ability to treat non isotropic turbulence was added by includ ing terms in the momentum equations which repro sent the deviations of the P y old stress gradients from those predi ted -- the standard k e model The form of the PANS equations then look like (SU\ _ SU\N ~ (_ 2 ~ phi— + U,y J = ~ ~P+3PkJ + P yea [(a + 2)
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From page 853... ...
, where there is little wwe generation The second case examined will be for Fr = Uc/(gd) ~/2 = 8 and Its = U d/~ = 12 700, where the w e action equal tion is solved for the unsteady turbulence generated wwes The surtacefiu tuation model in this case is 'one way' coupled, in the sense that the surfacer fluctuation variance is not 'fed back' into the free surface boundary conditions This sh ws that the w we z tion model produces reasonable result, when compared to su face fiuctutation measurements The final case is z two way coupled high Froud~number prediction, with Fr = 8 and file = 12 700 Here the near surface correction which produces the turbulent stress znisotropJ is disabled, but the gradient of the su face fluctuation variance is used for the free surface boundary conditions These two way coupled results are compared to experimental data for both the velocity field and the free surface In the sections that toll w, the discussion of ret suits will center on comparisons of transverse planes of experimental and computational results Compaq isons of computed and experimental results for sup face fluctuations in on the frss surface plans will also be presented For rsfsrsncs and to orient the reader, figure 2 show the predicted mean streamviss veloc ity U/U for the entire computational volume The vertical symmetry plans plans of the jet is shown, as are the far d wnstream exit plans for the volume, and the frs~surtacs plans Ths origin of the jet at the up stream inlet plans of the volume at x = 0 is clearly visible, as is the decay in velocity with streamvis dis tancs and intern tion with the f es surtacs 'Float ing'zbovs the volume in figure 2 is z plans sh wing the computed energy distribution :'2/dfor the from surface fluctuations 4.1 Modeling of a Low-FroudeNumber Jet The first case to be examined is z I w Froud~numbsr jet, e: Fr = 0 and file = 12 700 The f es sup face will he tzEsn to be fiat.
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From page 854... ...
, as well at the turbulence kinetic energy k and the dissi potion e, are required For n fully coupled approach, the resulting su face elevation variance would then be used in calculating the apparent stress acting at the free surface For n one w J coupled approach, the ret suits from the subsurface fl w are used in calculating the su face fluctuations, but the su face fluctuations are not used in calculating the sub su face flow in this approach, the sub surface fl w is calculated as suming the fl w is at zero F oude number In this case, the sub su face fl w behaves eon tlJ as that described in the previous se tion, and the only difference is that now, the surface fluctuations are cal culated, as well This was accomplished by advancing the w we n tion conservation equation in time in con junction the P Molds weraged equations The flow was initialized in n similar fashion as that used above, with the additional provision that the w we spectrum was initialized to zero at all locations The calcu lataions, again converged in 5000 time steps; sines the w Action equation in the form used hers is lin ear, it imposes no additional difltcultiss in obtaining a converged solution Figure 5 sh w the root mean square (r m s ) sup face fluctuation level z//d for the one way coupled computations (figure A)
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From page 855... ...
Again the may jor difference is in the cross flow velocity vectors; in the computations there is gain only n small outward fi w at the free surface The results in this section sh w that in n two way coupled approach, the magnitude of the r m s sup face fluctuations, and the location of the peak are reasonably well predicted the trenmvise e tent of the peak is over estimated, with the elevated region e tending too far downstream Even so, the spread ing of the jet near the free surface observed in the experimental data is not captured nccurntelJ This is mo t likely related to the high F oude number nag tore of the modeling being used The underlying as gumption of the modeling is that there is no near surface tress nnisotropJ, and the free so face eke ts are confined completely to the apparent stress ret suiting from the gradient in j2 This may be an over simplification of the problems, since the results of Hong & Walker (2000) sh w that, while the levels of nnisotropJ are smaller in high F oude number jets, anisotropJ become more impo tant with increasing strsamviss di tancs, and can still feet the beh A' :r of the fi w An sfisctivs, rational method for blending the high and low F ouds number approaches, turn ing on the anisotropJ e: appropriate, has yet to be developed 5 Summary and Conclusions The objective of this study w to develop an ap Broach to prsdi ting turbulent free surface flows in the coots t of the conventional PsJnolds aver Red N
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From page 856... ...
The predicted surtac~ fluctuntion distribution was in good ngreement with the observations FinallJ, n two WnJ coupled calcu Intion w presented, where the su face fluctuntions were used in the boundarJ conditions on the sub surtace flow This ngrees less well with the experi ments; the magnitude of the outward vefocitJ nt the free su face was substantinllJ under estimated This is believed to be due mninlJ to n Inck of nn efle~ tive method for switching between the I w Froud~ number nnd high F oude number trentment of the free surtace Acknowledgment This is work was supported bJ the Ofl ce of Naval P~ senrch under Contract Nos N00014 99 M 0082 nnd N00014 00 C 0057 monitored bJ Dr E P Pood References [1]
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From page 857... ...
f ~ l l l l lll 1 1 1 1 1 1 11, / / / ~~D .
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From page 858... ...
~ em- ~ ~ G ~ ~ ~1 ED ., Q 5 ~ ~ ~ ~ ~ A , 1'~: ~ O en, s.
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From page 861... ...
The spectral wave model is forcedby pressme fluctuations generated by the underlying turbu lent flow Ed the resulting a ~ era ed free surface topology is further modified by th near surface me m flow The turbule t somce term is extremely import mt it is obtained by as mmmg that the pressme spectrum is isohopic mdum3lteredbythe presence of the fiee surface The spechal wave model c mbe pled back to the me m flow solver vie the free surface elevation field Results are provided fu st for She low Frau de no jet with essentially zero fiee su face deformation Ed negligible wave dissipation These recohs show the utility of applying misotropy to the near-surface turbulence Ed illustrate She formation of c su face current The results also suggest that the layer over which the misotropy exists may be thim r that She model predicts Next, the model is applied to c high Froude no jet withbodh one-way mdtwo-way coupling Show in the fomm of free surface elevation, the recohs seem to suggest that the coupling is one way This is co mite mmitive Ed merits some further consideration The results were obtcinedby neglecting misotropy m the turbulence field, cssummg that since She flee su face is allow d to deform, turbulent fluctuations are uninhibited Ed misotropy carmot develop (use of m isotropic pressure spech um es c forcing function for the .. a ve model is consistent with f is assumption)
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From page 862... ...
ener en enoughtodist rbthefieesurface Indhese regions, he near-surface turbulence will be misoh opic md will affect the development of the me m flow
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