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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS 579 Large-Eddy Simulations of Turbulent Wake Flows Shaoping Shi, Andrei Smirnov, Ismail Celik (West Virginia University, USA) ABSTRACT A numerical method for solving three-dimensional, time-dependent incompressible Navier-Stokes equations using the large eddy simulation (LES) is briefly described. A new random flow generation technique (RFG) was used to provide the turbulent inflow boundary conditions. The combined LES-RFG procedure was applied to simulate the wake of flat plate and a ship model. The simulations of flat-plate wake were validated against the experimental data. In the case of a ship-wake (ship model 5415), reasonable results were obtained. 1 INTRODUCTION Most of the computational fluid dynamics efforts applied to flow past ships are based on Reynolds-Averaged Navier- Stokes (RANS) equations utilizing various turbulence models (Sotiropoulos and Patel, 1995; Paterson et al., 1996; Ratcliffe, 1998). The commonly used models include k-ε, k-ω and algebraic stress models. RANS is often quite adequate for mean flow predictions, but provides only limited information about turbulence characteristics and almost no details on large-scale unsteady structures of the flow field. LES technique, on the other hand, was designed to simulate the unsteady behavior of at least the large coherent turbulent eddies. There is hardly any study in the literature where LES is applied to flow around ship-hulls including the wake. The main reason obviously lies on the computer resource limitation which increases almost exponentially with the Reynolds number. In LES, the energy containing eddies of the flow are computed explicitly, while only the more universal (isotropic) small eddies are modeled. Thus very fine grids have to be applied in order to resolve the boundary layer near the wall where the turbulence length scales tend to zero as the the wall is approached. However, in some applications, like bubble dynamics modeling, it is still necessary to resolve coherent flow structures—large turbulent vortices and eddies. In RANS simulations, especially those using two-equation models such as k-ε model, these unsteady flow features are usually smeared out. The prediction of bubble population in the wake is important in controlling the signature of surface ships. Since it can be computationally prohibitive to include the ship hull and the wake in LES, it would be desirable for the purposes of pure wake simulations to start the computations somewhere in the near wake excluding the ship-body. This technique, called by other researchers the Initial Data Plane (IDP) approach (Hyman, 1998; Paterson et al., 1996; Dommermuth et al., 1996) can introduce large errors as pointed out by Hyman (1998). By applying this procedure the flow field in the far wake can be calculated. Then the wake computation can start form a plane in the near wake. This is similar to our approach in which we call it inflow boundary instead of IDP. In IDP method, the data reflects the mean flow, mean turbulence quantities and the scalar field at this plane. In our approach, a Random Flow Generation (RFG) algorithm (Smirnov et al., 2000b; Shi et al., 2000b) is applied to generate the inflow turbulence based on the information from experiments or RANS calculations. The appropriate time scale and length scale can be specified for each point at the inflow boundary also by using the RANS results. In addition to that all six turbulent shear stress components can be given, thus providing anisotropic and inhomogeneous turbulent inlet conditions. The generated velocity fluctuations satisfy instantaneous continuity equation, and the turbulence statistics (Reynolds the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS 581 it requires a very small time step due to numerical instability. Moreover, from our experience, this model shows high diffusion in a turbulent wake flow when used with relatively coarse grid. The original code has capability of handling curvilinear coordinates but here we only use Cartesian coordinates. As for numerics, Crank-Nicolson discretization scheme is applied for diagonal viscous and diffusive terms for time- advancement while Adams-Bashforth scheme was chosen for the other terms. Spatially, QUICK scheme (Leonard, 1979) or central differencing (CD) scheme is applied to discretize the convective terms. Central differencing scheme is used for the other terms. With QUICK scheme, no subgrid-scale (SGS) model is used while for central differencing scheme Smogarinsky model (Smagorinsky, 1963) is applied. More detailed information can be found in Zang, et al. (1994; 1993) and Shi et al. (2000a). 2.2 Random Flow Generation (RFG) methodology Here we provide a brief description of the method used for generating the inflow boundary. The algorithm proceeds in the following sequence. 1. Given an anisotropic velocity correlation tensor (say from a RANS) (8) of a turbulent flow field find an orthogonal transformation tensor aij that would diagonalize rij2 (9) (10) As a result of this step both aij and λn become known functions of space. 2. Generate a transient flow-field in a three-dimensional domain {vi(xj, t)}i-j−1..3 using the modified method of Kraichnan (?) (11) (12) (13) where l, τ are the length and time-scales of turbulence, εijk is the permutation tensor used in vector product operation (Spain, 1965), and N(M, σ) is a normal distribution with mean M and standard deviation σ. Symbols represent a sample of n wavenumber vectors and frequencies of the modeled turbulence spectrum (14) 3. Apply a scaling and orthogonal transformations to the flow-field vt generated in the previous step to obtain a new flow-field ui (15) (16) The procedure described above takes as an input the correlation tensor rij of the original turbulent flow-filed and information on length- and time-scales of turbulence (l, τ). The output of the procedure is the time dependent flow- field ui(xj, t), which satisfies the anisotropy and inhomogeneity of the original flow-field ũ i, i.e. the shear-stresses of the generated flow-field are equal to rij and length- and time-scales to l and τ respectively. As was shown by (Smirnov et al., 2000b; the authoritative version for attribution. 2 Repeated indexes imply summation, parentheses around indexes preclude summation.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS 582 Shi et al., 2000c), the new flow-field ui is divergence-free in case of a homogeneous turbulence and to a high-degree divergence free for an inhomogeneous turbulence. By virtue of Eq. (11), spatial and temporal variations of ui follow Gaussian distribution with characteristic length and time-scales of l, τ, however, other distributions can be used to simulate different turbulence spectra. More details of this method can be found in Smirnov et al. (2000b). 3 APPLICATIONS The pseudo-random flow field generated by the RFG method is added to the mean flow to established the boundary condition at the inlet plane. The overall procedure is applied to two significantly different cases, namely flat plate wake and a ship model wake. Since we try to avoid solving for the flow around the bodies, we start the computational domain from the inflow boundary (initial data plane) located immediately after the bodies in the wake where there is no flow- reversal. For the flat plate wake, we start the computational domain from 19.5mm (x/L=0.01) behind the edge of the plate, which corresponds to a location where measurements were available. For the ship wake the domain starts from x/L=1.05 (x/L=1. is the end of the ship model). At these planes the RFG method described above (Smirnov et al., 2000a) is used in conjunction with the the RANS calculations (Stern and Wilson, 2000). For both cases, the inflow and outflow boundary conditions are applied in x direction. For outflow boundary both convective and Neumann (free gradient) boundary conditions have been applied. A comparison (not shown here) indicated that there is not much difference between the results obtained by applying these two conditions. Symmetry conditions are used in y direction and periodic boundary conditions are used in the spanwise (z) direction. At the free surface slip in x and z directions is allowed but the velocity component normal to the free surface is set to zero. As such the free surface is approximated as a moving flat plane. For flat plate wake, the domain size is 1.0m×0.2m× 0.6m in x, y and z direction, respectively. The grid size is 82×50×50. Non-uniform grid is used in both x and y directions with stretching not exceeding 3 percent. Note that, in our study, x represents streamwise direction, y represents vertical direction and z represents spanwise direction, respectively (see Fig. 1). For the ship wake, the domain size is 1.5×0.2×0.6 (given in non-dimensional units in ship length) in x, y and z- directions, respectively. The grid size is 162× 50×66. Non-uniform grid is used in both x and y directions. The length scale and time scale are selected as constant in this paper. For the flat plate, the length scale is 4mm which is chosen as 10% of the width of the wake, and time scale is 0.001s. For ship wake, the length scale is 0.02, dimensionless of ship length, and the time scale is 0.01. 4 RESULTS 4.1 Flat Plate Wake In Fig. 2, the velocity time histories are shown at different points in the wake. From these pictures it is seen that both the amplitude and the frequency of the velocity decrease (decaying turbulence) along the streamwise direction, which is consistent with the behavior of a turbulent wake. The predicted anisotropy is also noteworthy. The logarithmic part of the mean velocity along streamwise direction at the center line can be expressed (Nakayama and Liu, 1990; Ramaprian and Patel, 1982; Wygnanski et al., 1986; Andreopoulos and Bradshaw, 1980) by (17) where Uc is the streamwise mean velocity at the center line, uτ=0.853m/s is the friction velocity of the boundary layer at the trailing edge, x is the distance from the plate edge, v is the kinetic viscosity and A and B are constants which are 4.65 and 0.7 respectively (Andreopoulos and Bradshaw, 1980). In the transverse direction (direction normal to the plate), the logarithmic part (if there exist one) of the mean velocity can be expressed as (Nakayama and Liu, 1990) the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS 583 (18) where U+≡U/uτ and y+≡yuτ/v. κ is the Von Karman constant and equal to 0.41. Figures 3 and 4 show the comparisons of our simulations with experimental results (Ramaprian and Patel, 1982). The predictions are in good agreement both with experiments and the theoretical log-profiles. The mean velocity along x- direction for our simulation has a wave-like oscillation. This phenomenon maybe due to a weak vortex shedding which could exist because we only performed the statistical analysis at one line instead of the whole center plane. Fig. 5 shows the decay of turbulent kinetic energy along the center line of the wake. We also depict the turbulent kinetic energy along a vertical line at x=381mm in Fig. 6. Again the predictions agree well with the experiments. All of the above results were obtained by using the QUICK scheme without any SGS model. This scheme has a built-in 4th order dissipation which seems to act like a SGS model. Large eddy simulations without SGS are possible as elucidated by Boris et al. (1992) The spanwise vorticity contours are shown in Fig. 7(a). Central differencing (CD) with Smagorinsky model was applied in this calculation. The results using the QUICK scheme were much smoother. In this Figure, large coherent structures are clearly visible. These structures are similar in appearance to Karman vortex street because they seem to be comprised of vortices of alternating sign of vorticity which are also visible in the experiments of Wygnanski et al. (1986). As explained by Wygnanski et al., “Neither the varicose mode, which requires that the vortices appear in pairs distributed symmetrically about the centerline, nor the sinuous mode, which requires vortices whose center coincides with the centerline, dominate this flow”. 4.2 SHIP WAKE The streamwise vorticity contours at different planes are shown in Fig. 8 and the contours of the vertical component of vorticity are shown in Fig. 9. Fig. 8 shows that concentrated vorticity decreases with axial distance, and in the far wake vorticity is only concentrated near the free surface. Some of this rapid decay may be partially due to the grid expanding towards the outlet plane, while the size of the wake increases in axial direction of the flow. As a result of this, many turbulence structures may have died out prematurely in simulations. It is interesting to note the two distinctly concentrated vorticity streaks away from the center line of the wake (Fig. 9). The streamwise velocity contours at a plane near the free surface are shown in Fig. 10. Due to the lack of experimental data for this wake, we can not make a quantitative judgment of the predictions. However, we think that these results are reasonable. As the width of the wake increases, turbulence decays. In the far wake only larger turbulence structures can been seen. This maybe due to two reasons: (a) coarser grids are applied in the far wake; (b) the small turbulence structures contain significantly less energy so that they can only last for shorter time compared to the larger turbulence structures. To test the effects of the grid size on the flow field resolution, we doubled the grid number in x (streamwise) direction and ran the simulations again. In Fig. 11 more detailed turbulence structures can been seen clearly as compared to the coarser grid solution (Fig. 9). The resolved turbulence kinetic energy (TKE) for different schemes and different grid sizes is shown in Fig. 12. The RANS solution is also shown in this picture for comparison. The resolved TKE from fine grid is higher than that of the coarse grid as expected. Central differencing discretization with Smagorinsky model gives better results than the other schemes. From this figure it also can be seen that there is not significant difference in the results when the QUICK scheme is used with or without SGS model. Moreover, the resolved TKE is lower than that of central differencing with Smagorinsky model. It means that QUICK scheme gives even higher numerical diffusion than the Smagorinsky model. This is mostly due to grid resolution. For more detailed information about the comparisons of numerical schemes and subgrid-scale models the reader is referred Shi et al. (2000a). One uncertainty in computations is presented by sinusoidal-like distribution of TKE in the near wake. It may be because of the existing surface wave (not accounted directively here). When the wave descends towards to the bottom of the domain, it seems to create a constriction with flow passing through a small area. Thus both the velocity and TKE are higher at this re the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS 584 gion. We checked the wave profile of RANS calculation and the peak region of k-profile does indeed corresponds to the descending wave. It is interesting that our calculations also provide a similar trend although no surface wave profile was applied in our LES. This indicates that some wave information may be present implicitly in the inflow boundary3. In figures 13–18 the velocity vectors on different vertical cross sectional planes are presented. Clearly, the large scale turbulent eddies (vortices) are captured, which can play an important role in bubble dynamics. The prediction of outward flow near free surface, and the streamwise evolution of the vortices are very encouraging indications of the success of the present LES approach. Currently, a case study with finer grids on the both vertical and spanwise direction is in progress. Here, we show the velocity vectors in a cross-sectional plane at x/L=0.2 (Fig. 15). Comparison with the results from a coarser grid in Fig. 13 shows that more turbulence eddies are captured. The streamwise vorticity contours on the same plane are also shown in Fig. 20 in oder to make the turbulent eddies more visible. The vorticity contours on different vertical planes shown in Fig. 19, Fig. 21 and Fig. 22 are indicative of the degree of resolution of the calculations. These structures are hard to see in Figures 13–18 due to relatively small magnitude of the velocity vectors. However, to demonstrate the small weak turbulence structures, like those in Fig. 16(a), we enlarged the velocity vectors where there is turbulence structure corresponding to the vorticity contour plot and depicted it in Fig. 16(b). A corresponding area in the vorticity contour plot is also included in Fig. 22 for reference. The cross section is at x/L=1.2. Having no interpolation, it seems that in the velocity vector plot the weak vortices can not be seen due to scale difference, whereas the interpolated contours plots do show small structures. Figures 21 and 22 show again that much of the vorticity is concentrated near the free surface and there are two large counter rotating vortices on two sides of the wake. 5 CONCLUSIONS AND FUTURE WORK A combined LES scheme and an RFG approach suitable for wake flows is briefly described. The wake flow of a flat plate is used as a validation case. Both the mean flow and turbulence intensities are compared with the experimental results. A good agreement has been obtained. Regarding ship-wake simulations it is difficult to draw any definite conclusions at this point in the absence of experimental data. However, the results are very reasonable and it means our approach is a viable approach in that it is capable of capturing the most energetic unsteady, turbulent vortices and/or eddies present in the wake. This study shows that with reasonable large grid nodes LES of ship wakes has very good prospects. Nevertheless, full scale LES calculations of ship flow are still impossible because of the limitation of computer resources and may require another 10 or 20 years (Larsson et al., 1998). According to Hyman (Hyman, 1998), free-surface/turbulence interaction is important for the near-surface lateral wake growth. So, including the effects of free surface will be the focus of our future work. Parallel computing (Osman et al., 2000) is also an important factor since it can facilitate the usage of much finer grid resolution in LES. ACKNOWLEDGEMENTS This work has been performed under a DOD EPSCoR project sponsored by the Office of Naval Research (ONR), Grant No. N000l4–98–1–0611. The program monitor is Dr. Edwin P.Rood. We thank Prof. Fred Stern and Dr. Robert Wilson of University of IOWA for providing the RANS results for the ship model. Special thanks are due to Prof. R.Street of Stanford University for providing us with the basic LES code. Prof. V.C.Patel prompt response with the report that contained the experimental data for the flat plate wake is also appreciated. the authoritative version for attribution. 3An animation of the ship wake flow is also available at http://cfd.mae.wvu.edu/shipwake

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS 585 REFERENCES Andreopoulos, J. and Bradshaw, P.: 1980, Measurements of interacting turbulent shear laysers in the near wake of a flat plate, J.Fluid Mech. 100, 639 Boris, J., Grinstein, F., Oran, E., and Kolbe, R.: 1992, New insights into large eddy simulation. Fluid Dynamics Research 10, 199 Dommermuth, D., Gharib, M., Huang, H., Innis, G., Maheo, P., Novikov, E., Talcott, J., and Wyaatt, D.: 1996, Turbulent free shear flows: A comparison between numerical simulations and experimental measurements, in 21st Symposium on Naval Hydrodynamics, pp 200–215, Trondhiem, Norway Hyman, M.: 1998, Computation of ship wake flows with free-surface/turbulence interaction, in 22nd Symposium on Naval Hydrodynamics, pp 11–32, Washington, D.C. Larsson, L., Regnstrom, B., Li, D., and Janson, C.: 1998, Failures, fantasies, and feats in the theoretical/numerical prediction of ship performance, in 22nd Symposium on Naval Hydrodynamics, pp 11–32, Washington, D.C. Leonard, B.: 1979, A stable and accurate convective modeling procedure based on quadratic upstream interpolation, Comput. Methods Appl. Mech. Engr. 19, 59 Nakayama, A. and Liu, B.: 1990, The turbulent near wake of a flat plate at low reynolds number, J.Fluid Mech. 217, 93 Osman, A., Ammar, H., Smirnov, A., Shi, S., and Celik, I.: 2000, Parallel analysis and implementation of large eddy simulations of ship wakes, in IEEE International Conference on Cluster Computing CLUSTER 2000 Technische Universitt Chemnitz, Saxony, Germany Paterson, E., Hyman, M., Stern, F., P., C., Bonetto, F., Drew, D., and Lahey, R.: 1996, Near- and far-field cfd for a naval combatant including thermal- stratification and two-fluid modeling, in 21st Symposium on Naval Hydrodynamics, pp 102–117, Trondheim, Norway Ramaprian, B. and Patel, V.: 1982, The symmetric turbulent wake of a flat plate, AIAA Journal 20, 1228 Ramaprian, B., Patel, V., and Sastry, M.: 1981, Turbulent wake development behind streamlined bodies, Technical Report IIHR Report No. 231, Iowa Institute of Hydraulic Research, The University of IOWA Ratcliffe, T.: 1998, Validation of free surface reynold's averaged navier stokes and potential flow codes, in 22nd Symposium on Naval Hydrodynamics, pp 964–980, Washington, D.C. Shi, S., Celik, I., and Smirnov, A.: 2000a, Comparison of different numerical schemes and sub-grid scale models in large-eddy simulation, in ASME Fluids Engineering Division Summer Meeting, No. 11232 in FEDSM2000, Boston, Massachusetts Shi, S., Smirnov, A., and Celik, I.: 2000b, An approach for generating time-dependent inflow boundary with application to large-eddy simulation of flat plate wake, in International Mechancial Engineering Congress and Exposition, Orlando, FL Shi, S., Smirnov, A., and Celik, I.: 2000c, Large eddy simulations of particle-laden turbulent wakes using a random flow generation technique, in ONR 2000 Free Surface Turbulence and Bubbly Flows Workshop, pp 13.1–13.7, California Institute of Technology, Pasadena, CA Smagorinsky, J.: 1963, General circulation experiments with the primitive equations, part i: the basic experiment, Monthly Weather Rev. 91, 99 Smirnov, A., Shi, S., and Celik, I.: 2000a, Random Flow Generation Technique in Large Eddy Simulations and Particle-Dynamics Modeling, submitted to Journal of Fluid Engineering Smirnov, A., Shi, S., and Celik, I.: 2000b, Random flow simulations with a bubble dynamics model, in ASME Fluids Engineering Division Summer Meeting, No. 11215 in FEDSM2000, Boston, Massachusetts Sotiropoulos, F. and Patel, V.: 1995, Application of reynolds-stress transport models of stern and wake flows, Journal of Ship Research 39, 263 Spain, B.: 1965, Tensor Calculus, Oliver and Boyd Stern, F. and Wilson, R.: 2000, RANS Results of Ship-Wake simulations, Private communication Wygnanski, I., Champagne, F., and Marasli, B.: 1986, On the large-scale structures in two-dimensional, small-deficit, turbulent wakes, J.Fluid Mech. 168, 31 Zang, Y., Street, R., and Koseff, J.: 1993, A dynamic the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS curvilinear coordinates. Journal of Computational Physics 114, 18 mixed subgrid-scale model and its applocation to turbulent recirculating flows, Phys. Fluids 5–12, 3186 586 Zang, Y., Street, R., and Koseff, J.: 1994, A non-staggered grid, fractional step method for time-dependent incompressible navier-stokes equations in

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. wake (x=0 is the inflow boundary) Figure 1: The schematic of the flat plate wake LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS Figure 2: Temporal history of the instantaneous velocity components at different points on the center line of the flat plate 587

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Ramaprian, et al. (1981) Figure 3: Centerline velocity in the flat plate wake LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS Figure 5: Kinetic energy profile along the center line in the wake of flat plate Figure 6: Kinetic energy profile at x=381mm (x/L= 0.2) in the wake of the flat plate Figure 4: U-component of the mean-velocity profile in the wake of the flat plate The experimental results are from 588

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS Figure 8: The streamwise vorticity (ωx)contours on different y-z plane in the ship wake Figure 9: The vorticity contours (ωy) on a x-z plane parallel to free surface at y/L=−0.01 in the ship wake 590

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS Figure 11: The same as Fig. 9 but double grid number in streamwise direction Figure 10: The streamwise velocity contours on a x-z plane parallel to free surface at y/L=−0.01 in the ship wake 591

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS Figure 14: The velocity vectors on y-z plane at x/L= 0.6 in the ship wake Figure 13: The velocity vectors on y-z plane at x/L= 0.2 in the ship wake Figure 12: Comparison of resolved turbulence kinetic energy for different computing cases Figure 15: The velocity vectors of finer grid solution on y-z plane at x/L=0.2 in the ship wake 592

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS Figure 18: The velocity vectors on y-z plane at x/L= 1.4 in the ship wake Figure 17: The velocity vectors on y-z plane at x/L= 1.0 in the ship wake Figure 16: The velocity vectors and the enlargement of area A on y-z plane at x/L=1.2 in the ship wake 593

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. directions) LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS Figure 22: The vorticity contours (ωx) on y-z plane at x/L=1.2 in the ship wake Figure 21: The vorticity contours (ωx) on y-z plane at x/L=0.6 in the ship wake Figure 19: The vorticity contours (ωx) on y-z plane at x/L=0.2 in the ship wake Figure 20: The vorticity contours (ωx) on y-z plane at x/L=0.2 in the ship wake (finer grids are applied on both y and z 594

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS 595 DISCUSSION M.Hyman Coastal Systems Station, USA This paper represents the next stage in an effort to obtain high resolution flow computations around and downstream of a surface ship. Many parabolic algorithms exist to compute wake flow and more recently, Dommermuth (1994) applied LES to the wake of a ship. That temporal simulation could, strictly speaking, begin some distance downstream of the stern at a location where the flow was decaying fairly slowly. The present work reports a spatial simulation which can, in theory, be applied very near the stern and extend as far downstream as computational resources permit. The method can be initiated by applying the authors' random flow generation (RFG) methodology. The RFG allows one to take RaNS simulation results and create a very plausible estimate of the instantaneous turbulent flow field. Once this flow field is available, a LES calculation of the near wake can be undertaken. The authors show several interesting results from applying their code to compute the wake of a flat plate and of a surface ship. While it is clear that this work is in its' early stages, there are several issues that should be considered in the course of future work. The near wake of a ship presumably contains a wide range of length scales from flow off the hull boundary layer, separated stern flow and propulsor flow. This complex flow is further complicated by breaking stern waves at higher Froude numbers. I would like to ask the authors how they would propose to characterize this flow by a single length scale, what this length scale should be and how this length scale should relate to grid resolution for LES calculations. In addition, I would like to ask if the authors' experience with LES calculations using the QUICK scheme (and lack of SGS model) could be related to grid resolution. AUTHOR'S REPLY We appreciate Dr. Hyman's summary and comments on this paper. What follows is a brief response to his questions. QUESTIONS Question 1: “I would like to ask the authors how they would propose to characterize this flow by a single length- scale, what this length-scale should be and how this length-scale should relate to grid resolution for LES calculations.” Answer. Though a constant was used in this study, in our method it is not necessary to use a single length-scale. In fact, in the current version the algorithm is already implemented with the length scale parameter as a function of space which can be computed from k and ε or k and ω. It adds practically no extra computational burden to the procedure. There can be a slight loss of accuracy in satisfying the instantaneous continuity equation in a flow field generated with the RFG procedure associated with using inhomogeneous turbulent length-scale. However, for the purposes of inlet plane initialization a small violation of continuity can be tolerated since the flow solver will automatically adjust the velocities in the next plane such that the flow field satisfies continuity. The length scale used in the RFG procedure is currently not related to grid resolution. This might be a good idea to pursue in the future. Question 2: I would like to ask if the authors' experience with LES calculations using the QUICK scheme (and lack of SGS model) could be related to grid resolution. Answer: From Fig. 12 in the paper, it can be seen that there is no significant difference in the results when QUICK scheme is used with or without SGS model. Moreover, the resolved turbulent kinetic energy is lower than that of central differencing with Smagorinsky model. This means that QUICK scheme gives even higher diffusion than Smagorinsky model. Theoretically, the numerical diffusion increases with increasing grid size, so the lower resolution of scale in the LES calculations using the QUICK scheme (and lack of SGS model) is most probably related to grid resolution. However, at this point we could not make assertion with regards to how strong the resolution depends on the grid size. A new run with a finer grid resolution is underway to investigate this issue. the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS 596 DISCUSSION L.Davidson Charmers Univ. of Technology, Sweden The paper presents LES of compressible wake flow. The method is applied to the wake after a flat plate and after a ship model. The computations are started downstream of the bodies. At the inlet plane, turbulent inlet boundary conditions are generated by specifying a turbulent spectrum; the procedure is called a “Random Flow Generation” (RFG) technique. The authors claim that the RFG technique is new. Are they aware of the method develop by the group at ECL [1] used in Aeroacoustics? How are the results presented in Figs. 3–6 compared to what can be achieved with RANS (either eddy-viscosity models or second-moment closure)? Maybe the authors can include some comments? On p. 4, col. 2, the authors comment upon the waviness of the velocity profile in Fig. 3. They say that this is because they have averaged only along the centerline instead of in the whole plane. This is probably correct, but the same effect would of course be achieved if they increased the time integration. The results show in Figs. 3–6 have been obtained with the QUICK scheme and no SGS model. The authors say that they also have carried out simulations with central differencing and the Smagorinsky model. What about these results? It would be interesting to see a comparison. A lot of contour plots are presented. I don't think this is very interesting. I would prefer to see more quantitative results. Furthermore, I'm surprised that in the wake flow behind the ship model no comparison is made with experiments. Why? Doesn't it exist any experimental data for the ship model wake? If so, why was it chosen in the first place? REFERENCES 1. W.Bechara and C.Bailly and P.Lafon, “Stochastic Approach to Noise Modelling for Free Turbulent Flow”, AIAA J., volume 32, Number 3, pp. 455– 463, 1994 L.Davidson Chalmers Univ. of Technology, Sweden 2. How are the results presented in Figs. 3–6 compared to what can be achieved with RANS (either eddy-viscosity models or second-moment closures)? 3. The results shown in Figs. 3–6 have been obtained with the QUICK scheme and no SGS model. The authors say that they also have carried out simulations with central differencing and the Smagorinsky model. What about these results? It would be interesting to see a comparison. 4. I'm surprised that in the wake flow behind the ship model, no comparison is made with experiments. Why? AUTHOR'S REPLY We would like to point out that there must be a typographical error in the first paragraph of reviewer's comments. This paper presents LES of incompressible rather than compressible flow. We agree with the reviewer that one can find numerous applications of Kraichnan's method pertaining to dispersed phase modeling, turbulence-generated noise etc. Our RFG technique is also based on the Kraichnan method. However, the original method was modified by us so as to handle the case of anisotropic turbulence while preserving the continuity of the flow field. Another novelty is the way the technique is implemented in conjunction with LES, and Lagrangian particle dynamics routines, which enable an efficient and flexible flow-field generation by matching all the Reynolds stress components measured or obtained from RANS (see for details Smirnov, et al. 2000). In this respect we do not claim RFG to be a new method, but rather a new technique used to augment the LES and particle dynamics computations. The comment of the reviewer about averaging may have resulted from some misunderstanding. Probably we did not make it clear enough in the paper that the velocity profile in Fig. 3 was not a result of space averaging along the centerline, but rather of time averaging. In this context there is no disagreement with the reviewer's statement. Answer to Question 2: Ramaprian, et al. (1981) performed RANS calculations with the k-ε model. Their results agree fairly well with their measurements. However, no matter what the the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS 597 outcome of this comparison may be, LES technique provides extra information on unsteady large-scale component of the flow-field, and thereby provides a more consistent and necessary approach for modeling Lagrangian bubble dynamics in the wake which is the ultimate goal of the present work. This, in our opinion, gives LES a definite advantage over RANS methods in this particular application area. Answer to Question 3: Fig. 1 included here shows the streamwise velocity at the center line calculated by using QUICK scheme without subgrid sale model and central differencing with Smagorinsky mode. As expect, the central differencing scheme gives better results compared to the QUICK scheme. This is also true for the turbulent intensities (not shown here). Answer to Question 4: Measurements of the flow-field in the wake of Navy ship model 5415 is an on-going work and the experimental results are not available at this time. In this respect our ship-wake results represent a prediction rather than a post-diction. Our results will be compared with experimental data as they become available. On the other hand our flat-plate computations presented in this paper were specifically done to validate our model and provide justification for extending its application to the ship-wake. It must be pointed out that the LES predictions were done for a shear free flat wall instead of wavy free surface. The influence of waves on turbulence is usually not negligible. the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. LARGE-EDDY SIMULATIONS OF TURBULENT WAKE FLOWS number 11215 in FEDSM2000, Boston, Massachusetts, 2000. REFERENCE: Fig. 1. Modified comparison of predicted mean streamwise velocity at center line with measurements. 1. A.Smirnov, S.Shi, and I.Celik. Random flow simulations with a bubble dynamics model. ASME Fluids Engineering Division Summer Meeting, 598