Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics (1994) / Chapter Skim
Currently Skimming:
Session 5- Wavy/Free Surface Flow: Viscous Flow and Internal Waves
Session 5- Wavy/Free Surface Flow: Viscous Flow and Internal Waves
Pages 213-310
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 215... ...
For laminar flow, the micro-scale flow indicates that the free-surface boundary conditions have a profound influence over the boundary layer and near and intermediate wake: the wave elevation and slopes correlate with the depthwise velocity; the streamwise and transverse velocities and vorticity display large variations, including islands of maximum/minimum values, whereas the depthwise velocity and pressure indicate small variations; significant free-surface vorticity flux and complex vorticity transport are displayed; wave-induced effects normalized by wave steepness are larger for small steepness with the exception of wave-induced separation; order-ofmagnitude estimates are confirmed; and appreciable errors are introduced through approximations to the free-surface boundary conditions. For turbulent flow, the results are similar, but preliminary due to the present uncertainty in appropriate treatment of the turbulence free-surface boundary conditions and meniscus boundary layer.
|
From page 216... ...
equations and boundary conditions for a solid-fluid juncture blw with waves. Some additional turbulent-flow solutions are also presented; however, these are preliminary due to the current uncertainty in prescribing appropriate turbulence free-surface boundary conditions and treatment of the meniscus boundary layer.
|
From page 217... ...
Any = (£ 1) Next, the normal and tangential dynamic and continuity-equation free-surface boundary conditions (see later)
|
From page 218... ...
, which facilitated the isolation and identification of the most important features of the wave-induced effects. Numerical results were presented for laminar and turbulent flow utilizing first-order boundary-layer equations and both smallamplitude-wave and more approximate zerogradient free-surface boundary conditions.
|
From page 219... ...
Note that 11 itself is unknown and must be determined as part of the solution. Boundary conditions are also required for the turbulence parameters (k,~.
|
From page 220... ...
and (5) are used to derive free-surface boundary conditions for V and p in conjunction with the solution for A
|
From page 221... ...
In summary, for laminar flow, the exact free-surface boundary conditions are given by (13)
|
From page 222... ...
In general, only detailed results are presented in which the exact freesurface boundary conditions were utilized; however, in the discussion of the error-bar charts, reference is made to the various approximate treatments discussed earlier. The discussion focuses on the differences between the Ak = 0 or equivalently deep solutions and the nonzero wave-steepness Ak = .01, .
|
From page 223... ...
Note that the solutions are for the primitive variables V and p subject to the free-surface boundary conditions (lab)
|
From page 224... ...
terms, i.e., teyWy + ozWz and the nature of Wy and Wz. The blw averaged values indicate that for the blw, the streamwise, transverse, and depthwise equations primarily represent balances between, respectively: convection, stretching, and diffusion terms 1, 2, 4, 5, 6, and 7; convection, stretching, and diffusion terms 1, 2, 5, 6, and 7; and convection and diffusion terms 1, 2, and 7.
|
From page 225... ...
Figures 1 la-d display AV contours and the blw averaged values of the dynamic and continuity-equation free-surface boundary conditions Uz, Vz, Wz, and p evaluated using (17~419)
|
From page 226... ...
, but the results indicate higher order. Noteworthy are the large values of transverse vorticity my for the blw, which are a direct consequence of the free-surface boundary conditions.
|
From page 227... ...
Also evident is that the waveinduced effects when normalized by Ak are larger for Ak = .01. The trends for the micro-scale flow are also similar to those described earlier for Ak = .01, except the following: there is an increased influence of satisfying the free-surface boundary conditions on the exact free-surface z = 1l; the wave elevation and slopes show complex behavior in the separation region, especially for Ak = .21; the depthwise velocity and pressure profiles indicate that the AU variations are reduced, whereas those for AV, AW, and Ap are increased; and the variations are confined to a narrower region close to the free surface.
|
From page 228... ...
The results for the error-bar chart for turbulent flow (figure 5) indicate that the differences between the errors for the zerogradient and turbulent-flow approximation to the exact free-surface boundary conditions are similar to those for laminar flow; however, it is also evident that the error has not been reduced to the same level.
|
From page 229... ...
For this purpose, current work involves further towing-tank experiments using laserdoppler velocimeter measurements for the foilplate model, including the region close to the free surface, and development of cfd methods, including moving contact-line boundary conditions and RaNS methods utilizing nonisotropic turbulence models and large-eddy and/or direct numerical simulations. Lastly, the present work has several implications with regard to the practical application of ship blw's for nonzero Fr.
|
From page 230... ...
14. Newman, J.N., "Recent Research on Ship Waves," Proc.
|
From page 231... ...
boundary layer (x y) free-surface Ifs ~ Y
|
From page 232... ...
11 1 01 SO Do so- nl- nln [O] L/ 19 o o~ouo00oooooooo.
|
From page 233... ...
N ~, ID ~o ~ -ail-: i:4 to ; a- ' ' ~' ~ 5 .
|
From page 234... ...
1 1 _ Net\ rot ~ \ A \ 408 so oos- soo l~qli&1 r r r I so 0 so 0 J`?
|
From page 235... ...
tD o ~c~ -- v- - ~ u, m o 0 000000OOO OU~000000Oo X oe~°~°oo |
From page 236... ...
Wave elevation and slopes: turbulent flow and Ak=.Ol. Figure 15.
|
From page 237... ...
ins.
|
From page 238... ...
The free-surface boundary conditions have been shown to have an important influence in regions of large velocity gradients and wave slopes. For practical applications, the boundary layer is relatively thick, especially over the afterbody and wake, such that the region of influence of the free-surface boundary conditions is expected to be large over a significant portion of the blw, which will effect the detailed flow near the free surface, i.e., velocity, pressure, and vorticity values and gradients, bubble entrainment, and triggering of wave breaking and induced separations, etc.; thereby, impacting the overall ship performance, including signatures and propeller-hull interaction.
|
From page 239... ...
239 NOMENCLATURE (X, y, a) : t: Re: Reynolds number Rig: Taylor Reynolds number Fr: Froude number We Weber number p: Density a: Surface tension g: Gravity v: Kinematic viscosity Cartesian coordinates Time Total velocity field Rotational velocity field Potential field Free-surface elevation Unit normal on free surface Total velocity components Rotational velocity components Total pressure Vortical component of pressure Atmospheric pressure Correlation coefficient A: Grid spacing SGS stress tensor Scattering or dissipation rate Filter operator Hi Hi A: A: ni (a, v, w)
|
From page 240... ...
The basic assumption of LES is that the small-scale motion is statistically more universal even though the large-scale motion may strongly depend on geometry, external forces, boundary conditions, and initial conditions. Successful implementations of LES methods include studies of homogeneous turbulence, turbulent channel flow, and weather prediction.
|
From page 241... ...
Note that U may contain a portion of the irrotational field depending on how the boundary conditions are defined. Based on this Helmholtz decomposition of the velocity field, define the total-pressure lI in terms of a vertical pressure P and an irrotational pressure as follows: H=p_ 0¢- 2V¢eV~-F2Z Here, the velocity and pressure terms are respectively normalized by uc and pUC2, where UC is the characteristic velocity and p is the density.
|
From page 242... ...
The two tangential stress conditions are essentially free-slip boundary conditions, and they can be derived by eliminating the potentialflow and shear-stress terms in Equation (16) of Dommermuth i64.
|
From page 243... ...
The confluence of boundary conditions that occur at the intersection of the free surface and the ship hull can be rigorously treater! without using ad hoc extrapolation schemes.
|
From page 244... ...
The anisotropic and isotropic portions of the SGS stresses are modeled separately for both the grid and test filters. Here, we illustrate the test filter approach for the anisotropic stresses: it [kk = ~` Cm ((Jo )
|
From page 245... ...
, the momentum equations (5) , and all the boundary conditions.
|
From page 246... ...
The numerical simulations of exact gravity-waves provide a test of the nonlinear formulation of the inviscid free-surface boundary conditions, and similarly the simulations of viscous Airy-waves test the formulation of the viscous stress conditions. The investigations of Triad interactions demonstrate the capability to simulate the nonTinear interactions of three-climensional gravitycapillary waves.
|
From page 247... ...
The transport theorem in conjunction with the divergence theorem may be used to simplify the resulting equation. Upon substitution of the exact free-surface boundary conditions, the following formula is clerived: dt (/v 2 2 /sO 9~¢ 2Fr2 i Based on the preceding formula for the conservation of energy, we define the accumulation of absolute error: Pa (t)
|
From page 248... ...
This corresponds to 80% of the limiting steepness for a gravity wave. The horizontal grid spacing is Ax, the time step is At, and the wave period is T
|
From page 249... ...
We consider a special case that illustrates the directional spreading of wave energy due to triad interactions: kit = (ki costly, ki sin (~)
|
From page 250... ...
A linear analysis of this flow leads to Rayleigh's stability equation subject to the kinematic and dynamic free-surface boundary conditions i384. Our current profile has the following general form: U(z)
|
From page 251... ...
A 1283 numerical simulation was run until the kinetic energy decayed to about 25% of its initial value. Then the velocity field was reseated so that 251
|
From page 252... ...
The energy density and the wavenumbers are normalized using Kolmogorov units. The numerical simulations include the results of 1283 psuedo-spectral simulations (labeled ~ and II)
|
From page 253... ...
Free slip boundary conditions are initially user} on the plane z = 0. If a surface wave is also present, then the subsurface-velocity field is periodically extended above the plane z= 0.
|
From page 254... ...
during the simulations (LES runs 2 & 3) for both the anisotropic and isotropic portions of the SGS stresses.
|
From page 255... ...
the work that the turbulence performs on the waves, we choose to use the DNS free-surface boundary conditions (see Equation 10) as if the boundary-layer is fully resolved.
|
From page 256... ...
Granted, these results are for a single real ization of a turbulent flow at a low Reynolds number, but they do not support the notion that the flow near the free surface behaves like two-dimensional turbulence. 7.2 E`ree-Surface Roughness We define roughness as the free-surface disturbance that is causer!
|
From page 257... ...
In our simulations the surface wave is initially propagating along the ~-axis, and then the interactions of the wave with the turbulence move wave energy off of the ~-axis. One measure of how much scattering has occurred in our numerical simulations is the amount of potential energy that has nonzero y-wavenumbers, which we denote as Es The operator Es also includes the effects of roughness, but its effect on Es is minimal because the total potential energy due to roughness is less than 0.2~ of the wave potential energy.
|
From page 258... ...
7.4 Turbulent Dissipation of Waves We define turbulent dissipation as the attenuation of surface waves due to interactions with subsurface turbulence. The total wave energy EW is Ew=2: 9~+2F2 / SO r SO We ./S to + 71x + by - 1)
|
From page 259... ...
We propose to overcome this problem by using the boundary-layer formulation of the free-surface boundary conditions that sin a private conversation, Prof. Milgram noted that he has since measured dissipation rates in another facility that are similar to his previous measurements.
|
From page 260... ...
(1963) The generation of capillary waves by steep gravity waves.
|
From page 261... ...
4 The adjustment procedure uses an exponential function (expt-b~2t2~) to slowly couple the free-surface waves with the subsurface turbulent flow.
|
From page 262... ...
The Fourier Theory (gravity) amplitudes of the gravity waves (at = a2)
|
From page 264... ...
(b) LES and DNS runs are wave energy spectra at time t = 0.36.
|
From page 265... ...
The initial conditions for the total-velocity spectrum are shown. The initial wave energy is zero, corresponding to no wave.
|
From page 267... ...
The plots are based on DNS Run 4.
|
From page 268... ...
Es is l~he scattered potential wave energy and Ep is the initial potential wave energy.
|
From page 269... ...
ID it a' it · - / it a; it · - ~ LL - - ~ Fog_ ~ __ r ;~ hi_ i_ A brim it- - ..
|
From page 270... ...
A Helmholtz formulation also allows the use of boundary -layer approximations in the freesurface boundary conditions. This prevents the excessive dissipation of waves that is observed in RANS simulations of free-surface flows.
|
From page 271... ...
In the latter part, the flow characteristics of 3-D submerged foil is investigated Flows of five different cases are simulated and compare each other to discuss the free-surface effect on the hydrofoils where the submergence depth and angle of attack are imposed. Though the numerical simulation, it is found that the lift and drag are seriously influenced by the E-surface waves.
|
From page 272... ...
NUMERICAL SIMt] LATION OF SHIP WAVES Basic Equation Numerical simulation of 3-D iree-surface flow is carried out by solving the N-S equation basically following to the MAC method The mean velocity components u, v and w exclusive of the fluctuation components at (n+l)
|
From page 273... ...
where x is in the uniform flow direction, y in the lateral, and z in the vertical direction respectively; u, v and w are the velocity components in the x-, y- and z direction, respectively. They are nonnalized by the representative length and reference velocity.
|
From page 274... ...
is appro~mated as K'`n' because it is not easy to get the solution at ~e (n+l) time step.
|
From page 275... ...
With the wall condition, the minimum grid spacing in the direction normal to the body surface should be small enough to resolve the viscous sublayer in the boundary layer.
|
From page 276... ...
It seems not so easy to calculate the cross flows accurately at the stern part considenug some aspects in the nu~rmical point of views. The assumption of symmetry or the steady flows should be pointed out The unsteadiness and non-symmetry observed in experiment should be taken into account in the numerical simulation.
|
From page 277... ...
That of case 2 is much more intensive than that of case 3; it means that the velocity fields around the tip may be influenced by the free-surface waves. In a shallow depth of case 3, 1 he y-component velocity is reduced in magnitude entirely around He wing surface.
|
From page 278... ...
The computed free-surface waves are compared with those by the explicit method and experirrEnts. The CPU Are is reduced to less An half because each line of block pent~iagonal system in any of the thme sweeps is decoupled completely from alters and the highly vectorized 278
|
From page 279... ...
The no-slip condition used for the viscous flow simulation realizes the shock-free condition for the pressure around the hailing edge automatically, which proves to be one of the merits of the present numerical simulation. The la~e-amplitude waves, under the suspect of breaking, have much influence on the lift and drag force because the pressure and velocity gradients are seriously affected by the waves.
|
From page 280... ...
oooooo exp~iment(3.59xI06) Fig.5 Free-surface wave profile at hull surface
|
From page 281... ...
- ..... Fig.6 Velocity vectors on transverse sections at Fn=0.316 (a)
|
From page 282... ...
(c) =4 44H ~ _ _ _ = = 1 o.o Fig.10 Free-surface profile and velocity vectors on Me three parallel surfaces of case 2 (a)
|
From page 283... ...
05 (b) Fig.14 Pressure contour on wing surface for case 4 (a)
|
From page 284... ...
plane) TE l' l -- .50 Fig.20 Comparison of spanwise pressure distribudon on wing surface.
|
From page 285... ...
.. -= TE 1 if - -.50 Fig.24 Compatison of spanwise pressure distribution on wing surface Lower;suchon, upper;pressure ;case 4, - - - ;case5)
|
From page 286... ...
One of the numerical inconsistencies may be the different treatment of boundary conditions near the hull surface. The explicit method has a very simple boundary condition by nature inside the hull surface.
|
From page 287... ...
A five-point central differencings are used to increase the solution accuracy with a given number of grid points, thus resulting in block pentadiagonal systems. At points adjacent to the point of singularity or solid wall boundary, the three-point differencing formula are used for the first and second derivatives.
|
From page 289... ...
ABSTRACT Here we develop a non-linear theory for the solution of the near and intermediate internal wave field caused by a ship passing through a stratified ocean. The theory is mainly composed of a perturbation cross flow theory and is asymptotic to the case of very high supercritical ship speeds in the regime of greatest interest for modern ships, and to the case of small density differences.
|
From page 290... ...
It would therefore be desireable to have available a theoretical method for the prediction of ship internal waves including non-linear terms. In the present work we take advantage of the strongly divergent nature of the wave pattern for Fh>>1 to develop a numerical calculation method for the theoretical prediction of both near and far field internal wave patterns for arbitrary vertical density distributions and for slender ships of arbimary cross section, including the dominant non linear terms.
|
From page 291... ...
is assumed two-dimensional in the cross flow plane, and becomes the entire disturbance in the far field. We consider the generation and propagation of these ship internal waves in a frame of reference stationary relative to the ship.
|
From page 292... ...
Then, the pressure and density gradients interact to create vorticity in the region of stratification. This vorticity first creates a narrow region of displacements within the wake behind the ship, and these displacements relax to form internal waves propagating transversely.
|
From page 293... ...
(7) , we deduce the Poisson Equation in nondimensional form, which governs the generation and oscillation of the internal waves in the cross flow plane, with the vorticity as its forcing term, where, am + am = ~, ~(14)
|
From page 294... ...
In the numerical simulation however, they can be decoupled by an iteration technique or simply by separating their calculation by an infinitesimal time interval. We choose the latter since the internal waves are long waves of steepness around one tenth and the velocity field changes very slowly.
|
From page 295... ...
As discussed in the section above, velocity and vorticity fields can be solved separately at infinitesimal time intervals, or infinitesimal distances in the x direction. At the begining cross plane, x = xO, the vorticity is zero, the velocity field is taken from the homogeneous flow, us; the pycnoclines are horizontal with an imposed density profile, and then the first vorticity increment can be calculated, using Eq.~12)
|
From page 296... ...
So the accurate approximation of first and second order derivatives of the stream function is crucial in the numerical simulation of ship generated internal waves. We choose a high order Hermite element, because it has both the variable and its first order partial derivative all as unknowns, and allows tl~e second order derivatives, which do not occur in the governing equation, to be differentiated based on the velocity field which is interpolated by a linear combination of its discrete nodal values.
|
From page 297... ...
The cubic Hermite shape functions Ni and He bilinear shape functions Hi, and their first order partial derivatives aNi /at, aNi /arl and aHi /at, DH i /~ are available in the literature. With careful geometrical treatment of the physical domain and application of standard procedures in FEM, the stream function and velocity field can be solved together with sufficient accuracy.
|
From page 298... ...
RESULTS & DISCUSSION The numerical code developed based on the above theory and computational techniques has been tested with regard both to the finite element solution of the Poisson equation, and the convergence of wave amplitudes for different marching steps in the x direction. As a test flow, we consider a simple closed flow with a stream function, where, ~ = sinewy)
|
From page 299... ...
- A Ax=0.005 ~ .7 ~ ~ ~c=0.01 . ~L~_ -0 1 2 3 4 y/D 5 Vorticity within Pycnoclines: midship Figure 5: Convergence Test for Marching Sums in the x Direction, /`x.
|
From page 300... ...
/~ <1910) 1 ~O Figure 6: Ship Internal Waves, Case 41(c)
|
From page 301... ...
that the far field internal wave pattern may be calculated upon the assumption that the far field wave pattern originates from the initial conditions represented in the triple lobed pattern. In that case, they showed that the entire far field may be represented by a complex amplitude function, which is readily calculated from a Fourier transform of the triple lobed pattem, amplitudes and velocities.
|
From page 302... ...
- , Crests, - . -, Troughs, Bo~ from Direct Numerical Calculadons; ·~ , Crests and Troughs, from Far Field Theory.
|
From page 303... ...
A kin-map is shown in figure 11, in the case of 41b, which gives a clear view of the distribution of wave lengths in the patterns, and is helpful when far field wave amplitudes are analyzed using the amplitude function. The CPU time needed for calculation in the near field to the triple-lobe pattern is only around 265 seconds, and for far field calculation to 80 ship lengths in case 41c, 371 minutes.
|
From page 304... ...
: Longitudinal Wave Cuts at Transverse Distances ilom the Ship, Case 41c, hom y/D =4.35 to y/D = 40.35 in intervals of Ay/D = 4.5, Where , Experimental Data, - · -, Direct Calculation Using Current Theory, , Results From Far Field Theory Using Triple-Lobe Pattern From Direct Calculabon 304
|
From page 305... ...
1 1 1 1 1 1 1 _ ~ = ~ - ~ 1 1 1 1 1 1 1 = - - 3 -1 1, 1 1 1 1 1 1 1 1 1 1 1, I I I I I I I I I I I I I I I ~ i I I I I I I I I I I ~ = ~ i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I l 1 1 1 111111111,,,,I,,,,I,,,,_ -r T I I I l l I I 1 1 _ _ ~[ 1 1 1 1 1 :~ ~[ 1 1 1 1 : -1, , , , I , , , , I , , , , I , , , , I , , , , I , , , , I , , l , , , ~ I ~ ~ I ~ I ~ ~ ~ ~ r I ~ I ' I ~ ~ ' ' I ' ~ , I I , , I l_ 0.05 o.oo -0.05 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I l 1 I I I 1 1 , , , , 1 , , I 1 , , , , 1 , , , , 1 , , , , 1 , , , ,-o 10 20 30 40 50 60 7 x/L Figure lO |
From page 306... ...
: Longim~inal Wave Cuts at Transverse Distances from ~e Ship, Case 2Sb, from y/D= 5 to y/D= 32 in intervals of /ty/D= 4.5, Where , Experimental Data, ~ , Results From Far Field Theory Using Triple-Lobe Pattern From Direct Calculadon 306
|
From page 307... ...
: Longi~dinal Wave Cuts at Transverse Distances from dle Ship, Case 25c from y/D= 5 to y/D= 32 in intervals of Ay/D = 4.5, Where , Experimental Data, ·~ , Results From Far Field Theory Using Triple-Lobe Pattern From Direct Calculabon 307
|
From page 308... ...
The resulting solution satisfies the boundary conditions everywhere on the ship hull. The vorticity is calculated by a marching procedure, using an algorithm based on Fridman's Equation.
|
From page 309... ...
"Ship Internal Waves in a Shallow Thermocline: the Supersonic Case", Proc. of the 1 8th Symposium on Naval Hydrodynamics, National Academy Press, 1990 Wang, H
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.