S.Jordan (Naval Undersea Warfare Center, USA)

The present investigation discusses the resolution of the turbulent vortical motion behind two bluff bodies. The LES results of the cylinder wake at Reynolds number of 5,600 showed good comparisons to the published experimental data in terms of the global and local wake characteristics such as the drag and base pressure coefficients, shedding frequencies, near wake structure, and the Reynolds stresses. Qualitatively, the time-averaged Reynolds stresses of the formation region revealed similar symmetric characteristics over the range 525 ≤Re≤140,000. The NACA 0018 hydrofoil simulation was performed at a Reynolds number of 35,000 and a zero angle-of-attack. Although the instantaneous flow is massively separated over most of the trailing edge (by evidence of the formed large-scale vortical structures), it does not transition to turbulence until near the tip.

Most fluid dynamic problems in the Navy involve very complex behaviors that cannot be accurately simulated using simple computational methods. Generally, the geometric topologies are quite arbitrary and require special consideration when attempting to predict the associated flow. The flow itself is commonly unsteady, incompressible and very turbulent. An excellent example of these characteristics is the turbulent wake behind a complex bluff body. Comprehensive studies of the vortical formations and the subsequent transport of the structures downstream can provide many important contributions related directly to Naval applications.

Numerical predictions of the turbulent wake characteristics poses an excellent challenge for the large-eddy simulation (LES). Unlike the full-scale modeling inherent in a Reynolds-Averaged Navier-Stokes (RANS) technique, the LES method requires resolution of the dominate energy-bearing scales of the turbulent field while modeling only the remaining finer eddies which tend toward homogeneous and isotropic characteristics. Separation of the resolved and modeled scales is established by spatially filtering the basic governing equations of the full fluid motion. In most computations however, this filter is actually treated implicitly through the spatial resolution of the implemented grid. Those physics lying beneath the grid’s resolution represent the subgrid scales (SGS) of the turbulent field. By design, the SGS model usually encompasses most of the equilibrium range of the turbulent kinetic energy field. Consequently, these models are much simpler in form than those developed for RANS computations and better delineate the turbulent physics of their assigned scales.

In the following paper, results from a LES computation will be presented that will describe the organized vortex motion in the near wake of the circular cylinder (see Fig. 1). Previous studies have shown that the primary Strouhal vortices derive most of their large-scale vorticity from the separated shear layers, and that they organize downstream to form the well-known Karman vortex street. Except for

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Advancements of the Large-Eddy Simulation for Turbulent Flow Over Complex Bluff Bodies
S.Jordan (Naval Undersea Warfare Center, USA)
ABSTRACT
The present investigation discusses the resolution of the turbulent vortical motion behind two bluff bodies. The LES results of the cylinder wake at Reynolds number of 5,600 showed good comparisons to the published experimental data in terms of the global and local wake characteristics such as the drag and base pressure coefficients, shedding frequencies, near wake structure, and the Reynolds stresses. Qualitatively, the time-averaged Reynolds stresses of the formation region revealed similar symmetric characteristics over the range 525 ≤Re≤140,000. The NACA 0018 hydrofoil simulation was performed at a Reynolds number of 35,000 and a zero angle-of-attack. Although the instantaneous flow is massively separated over most of the trailing edge (by evidence of the formed large-scale vortical structures), it does not transition to turbulence until near the tip.
1. INTRODUCTION
Most fluid dynamic problems in the Navy involve very complex behaviors that cannot be accurately simulated using simple computational methods. Generally, the geometric topologies are quite arbitrary and require special consideration when attempting to predict the associated flow. The flow itself is commonly unsteady, incompressible and very turbulent. An excellent example of these characteristics is the turbulent wake behind a complex bluff body. Comprehensive studies of the vortical formations and the subsequent transport of the structures downstream can provide many important contributions related directly to Naval applications.
Numerical predictions of the turbulent wake characteristics poses an excellent challenge for the large-eddy simulation (LES). Unlike the full-scale modeling inherent in a Reynolds-Averaged Navier-Stokes (RANS) technique, the LES method requires resolution of the dominate energy-bearing scales of the turbulent field while modeling only the remaining finer eddies which tend toward homogeneous and isotropic characteristics. Separation of the resolved and modeled scales is established by spatially filtering the basic governing equations of the full fluid motion. In most computations however, this filter is actually treated implicitly through the spatial resolution of the implemented grid. Those physics lying beneath the grid’s resolution represent the subgrid scales (SGS) of the turbulent field. By design, the SGS model usually encompasses most of the equilibrium range of the turbulent kinetic energy field. Consequently, these models are much simpler in form than those developed for RANS computations and better delineate the turbulent physics of their assigned scales.
In the following paper, results from a LES computation will be presented that will describe the organized vortex motion in the near wake of the circular cylinder (see Fig. 1). Previous studies have shown that the primary Strouhal vortices derive most of their large-scale vorticity from the separated shear layers, and that they organize downstream to form the well-known Karman vortex street. Except for

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only minor dispersion, the large scale motion remains strongly coherent for many characteristic lengths downstream of the cylinder. Closer to the cylinder, the upper and lower separated shear layers constitute the transverse outer regions of the formation regime. These layers define the transition activity between the point of laminar separation and the initial formation of the turbulent vortices. Besides these shear layers, the fluctuating base pressure near the downstream stagnant region, directly behind the cylinder, strongly influences the large-scale vortex characteristics.
The near wake remains fully turbulent over a many diameters downstream. Our experimental evidence suggests that the shed vorticies of the Karmon street display little variation in their cross-sectional area. Zhou and Antonia (1993) found for moderate Reynolds numbers (Re) that the convection velocity slowly increased downstream to over 90% of the freestream velocity after 50 diameters. While the peak vorticity as well as the magnitudes of the Reynolds stresses gradual decay downstream, the respective statistical distributions remain essentially unchanged. The peak horizontal stress components occur at the vortex center, but the circumferential intensities closely mimic that of an Oseen vortex.
Three LES investigations of the cylinder near wake flow have been formally published above the low-Re regime. The first was a study by Kato et al. (1993) who concentrated on predicting the aerodynamic noise in the near wake using the finite element method. Due to their highly dissipative subgrid-scale turbulence model and their relatively coarse mesh of the wake region, agreement with the experimental data in terms of the spectral physics was obtained only at the very low frequency levels. A study by Mittal and Moin (1997) focused on contrasting the predictive accuracy of second-order central and fifth-order upwind-biased schemes for the convective terms of the governing equations. Both schemes gave similar results provided a 20–30 percent finer grid was generated when selecting the lower order scheme. Using a curvilinear coordinate form of the basic LES formulation and dynamic SGS model, Jordan and Ragab (1998) showed excellent comparisons to the published experimental data in terms of both the global and local wake characteristics; such as the drag and base pressure coefficients, shedding and detection frequencies, peak vorticity, and the downstream mean velocity-defect and Reynolds stresses.
Figure 1: Large-Scale Physics of the Cylinder Near Wake.
The present paper will also discuss LES results of the turbulent wake motion resulting from trailing edge separation of the boundary layer of a NACA 0018 hydrofoil. The US Navy began utilizing the symmetric NACA 0018 section in 1970 to serve as their submarine control surfaces. This particular section as well as other symmetric thick sections can deliver high lift and low drag while maintaining strong structural integrity during complicated maneuvers. Thus, understanding both the instantaneous and mean character of the associated flow at various upstream conditions is crucial for effective use of these sections.
This material is the first in-depth numerical investigation of the fine-scale physics of the NACA 0018 section. The results were generated for upstream velocities at zero angle-of-attack. Previous numerical studies, notably by Hodge et al. (1978) and Sugavanam and Wu, (1980) (among others), used various phenomenological models to represent all the turbulent scales, thus their results can not provide useful details of the fine-scale physics.
To adequately resolve the turbulent wake flow of the cylinder and hydrofoil geometries, nonorthogonal O-type and C-type grid topologies were generated to facilitate control over the spatial resolution. The corresponding LES governing equations were solved in a generalized curvilinear coordinate framework. The dynamic SGS model of Germano et al. (1991) was reformulated for application to the curvilinear space. Since nonorthogonal topologies are often necessary to properly resolve the flow characteristics in many complex domains like these wake flows, the present formulation has extensive applicability.

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2. FORMULATION AND NUMERICAL METHOD
To derive a LES system applicable for complex domains, we begin with the Cartesian form comprised of the continuity and the Navier-Stokes equations. Extension of this coupled system to nonorthogonal grid topologies requires two operations; the transformation phase and the filtering phase. In the following approach, the initial system will be transformed prior to its filtering. This order-of-operations is chosen to facilitate the filter operation which should be administered along the curvilinear grid lines. The filter operates on both the flow quantity and the metric coefficient. Depicting the metric coefficients as filtered is justified herein because the finite-difference expressions used for approximating each metric coefficient are themselves separate mechanisms of spatial filtering. For the present, we will assume that the filter width and local grid spacing are equal; thus, the resolved and filtered turbulent fields are mathematically the same. The LES governing equations appear as
(1)
(2)
where each term is shown in its non-dimensional strong conservation-law form. The convective term is defined in terms of the resolvable contravariant velocity components . The coefficients and denote the filtered metrics and the filtered Jacobian of the transformation, respectively. Under this derivation, the subgrid scale (SGS) stress tensor is defined as .
To resolve the cylinder and hydrofoil wake flows, the above LES system was time-advanced by a variant of the fractional-step method (Jordan and Ragab, 1996). Third-order upwind-biased differences approximate the convective derivatives in the wake streamwise and transverse directions. The periodic spanwise components are approximated by a fourth-order accurate compact scheme. A Runge-Kutta procedure is used to time-advance the flow because of its strong numerical stability even under inviscid conditions. The viscous terms are time-split by the Crank-Nicolson scheme and spatially approximated by conservative finite-volume differences. The overall accuracy of the solutions are second order in both space and time. Through extensive testing of this fractional-step method for the wake flows, an error tolerance (residual) of 10−4 was found to be acceptable for terminating convergence of the pressure-Poisson equation. Under this criteria, incompressibility was reached usually in less than 100 iterations at each time increment. Further details of the solution methodology, along with several test cases, can be found in Jordan (1996) or Jordan and Ragab (1996).
3. DYNAMIC SUBGRID SCALE MODEL
The SGS field was modeled by Smagorinsky’s eddy viscosity relationship (Smagorinsky, 1963) and subsequently modified according to Germano et al. (1991) for dynamic computation of the model coefficient. This model gives the correct asymptotic behavior of the turbulent stresses when approaching solid walls and can suitably distinguish between laminar and turbulent flow regimes. For the present application, the SGS model was transformed to the computational space. The curvilinear form of the dynamic model appears as (Jordan and Ragab, 1998)
(3)
where C is considered as Smagorinsky’s coefficient and the filtered metric term is defined as . In this model, the turbulent eddy viscosity is defined as where and is the grid-filter width. The filtered strain-rate tensor is defined by
(4)
The contravariant form of the resolvable strain-rate field can be expressed as . By combining this transformation with equation (4), the complete definition of the tensor is

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(5a)
(5b)
The first term in equation (5a) is a direct contribution of the CDM to the transformed molecular diffusion term in equation (1), whereas the second term represents the contravariant components of the cell flux vector gradients. To maintain second-order accuracy in time, the Crank-Nicolson and Adams-Bashforth schemes were applied to the first and second components of the total viscous term, respectively.
4. MODEL COEFFICIENT
Smagorinsky’s coefficient in the CDM was acquired by implementing the procedures of Germano et al. (1991) and Lilly (1992). The governing LES equations in curvilinear coordinates were filtered a second time (by a test filter). This third operation gave two resolvable tensors; a modified Reynolds stress tensor
(6)
and a modified Leonard tensor
(7)
The second overbar indicates the test filter operation. In both these tensors, the components are computed by test filtering the Cartesian and the contravariant velocity components of the resolved field. As a result, an identity for the Leonard term arises that is defined as . Modeling the Reynolds stress tensor as
(8)
and subsequently using the identity for the Leonard term,
(9a)
(9b)
completes the SGS model definition in curvilinear coordinates. The filter width ratio is defined . Following the procedure of Lilly (1992), the CDM coefficient can be computed uniquely as
(10)
This model coefficient can yield both positive and negative values which depict forward and back scatter along the energy cascade, respectively. To inhibit the potential for diverging solutions, backscatter effects of the CDM were truncated to zero in the present applications. Also, a filter width ratio of α=2 was used based on the numerical testing reported by Jordan (1996). Finally, all explicit filtering in the wake simulations were conducted in the computational space using a box-type filter.
5. RESULTS AND DISCUSSION
The present LES investigations focus on the turbulent wake statistics of a circular cylinder and a NACA 0018 hydrofoil. Reynolds numbers for the cylinder and hydrofoil were Re=5,600 and Re=35,000 based on the diameter and cord length, respectively. In both simulations, the flow was impulsively started with unit velocity, zero reference pressure and a fixed non-dimensional time step. Once the lift and drag profiles indicated initiation of the shedding process,new velocity and pressure conditions were imposed along the exit flow boundary. The formulated exit conditions were transformed forms of the continuity and Euler equations, which were found to be satisfactory to exit the shed vortices. An associated exit pressure gradient was computed via the velocity update equation of the fractional-step method to serve as a Neuman boundary condition for the pressure field solution.
5.1 Circular Cylinder
The structured grid for the cylinder test case, shown in Fig. 2, was 241×241×32 (x,y,z directions). While the inflow and outflow outer boundaries were established at 10 and 20 diameters, respectively, the inner no-slip boundary represented the cylinder periphery (s). Along the periphery, the distribution of points was within the upstream

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Figure 2: Geometric and Flow Conditions of the Cylinder Near Wake at Re=5,600.
Figure 3. Comparison of the Computed Mean Pressure Coefficient to the Experimental Results of Norberg (1992); (3D)16 Depicts 16 Spanwise Points over Length π and (3D)32 Denotes 32 Points over Spanwise Length 2/3π.
laminar boundary layer and in the immediate formation region. To insure adequate resolution of the turbulent wall layers ( for the first point), all circumferential lines were clustered toward the cylinder surface. The spanwise spacing was based on the empirical relationship λz/D~20Re−1/2 for the scale (λz) of the large eddies. Finally, the spanwise end conditions were treated as periodic.
The base pressure coefficient is an important parameter to properly resolve because it correlates strongly with the formation length, Strouhal number and strength of the primary vortices. Evidence indicating its numerical accuracy in the present simulation is illustrated in Fig. 3; (Cp)b was time-averaged over six shedding cycles.
Figure 4. Phase-averaged SGS Model Results; (a) νT/ν Contours, Max. 0.2, Min. −0.2, and Incr. 0.02, (b) SGS/Dm through Wake Centerline.
LES results using only 16 points over spanwise length π as well as a 2D computation clearly illustrate the gross errors attained after separation when the three-dimensionality of the wake flow is ignored or the grid contains a poor spanwise resolution. With 32 points, the drag coefficient averaged to approximately <CD>=1.01 which is within 2 percent of the experimental value found in White (1974)
Before presenting the LES results of the near wake region, questions regarding significant contributions from SGS model must be addressed relative to the inherent artificial dissipation produced when using third-order upwind-biased differences. To appreciate the model’s quality, the investigation should be local. Figure 4 shows phase-averaged results with homogeneity assumed in the spanwise direction. The νT contours (scaled by ν) illustrate negligible contributions in the laminar flow regions and in the wake formation region where the grid

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resolution is the highest. Throughout the near wake, nearly 10 percent of the values are greater than 0.1 with approximately 6 percent being positive. Streamwise contributions (SGS)u along the wake centerline (scaled by Dm) suggest a dominant role played by the model locally in regions of high strain-rate activity. Like the vT contours, only slightly more than one-half of these scaled terms were positive.
Figure 5. Typical Snapshot of the Three-dimensionality of the Cylinder Near Wake; Iso-surfaces of Vorticity Magnitude (ω=2) Revealing Streamwise “Fingers”.
The snapshot in Fig. 5 of the near wake, in terms of the magnitude of vorticity (ω), clearly displays the formation region structure, Strouhal vortices, and the large-scale streamwise structures connecting the alternating shed vortices. The latter elongated filaments have been experimentally viewed and termed as “fingers” (Gerrard, 1978). These intermediate structures of the near wake persist downstream and posses a high degree of streamwise vorticity, but low streamwise velocity. Thus, in terms of the helicity quantity they are on the same local scale as the adjacent Strouhal vortices which by comparison contain a high degree of spanwise vorticity, but low spanwise velocity. Each finger is comprised of a pair of counter-rotating vortices with spanwise separation lengths of approximately one diameter (Bays-Muchmore and Ahmed, 1993).
Past experimental observations and measurements conclude that the most fundamental physics of Strouhal vortices originate from the formation region. The turbulent statistics of that region (time-averaged over six shedding cycles and spatially averaged in the spanwise direction) are plotted in Fig. 6 in terms of the normalized total
Figure 6. Reynolds Stress Statistics within the Formation Region; (a) Contour Max. 0.2. Incr. 0.02, (b) Contour Max. 0.42, Incr. 0.042 and (c) Contour Max. 0.12, Min. −0.12, Incr. 0.012. Contours Time-averaged over Six Shedding Cycles.
Reynolds stresses and are the periodic and random components, respectively. The figure shows negligible levels of total stress within the upstream laminar regimes and along the cylinder periphery. Within the separated shear layers, the sharp streamwise contours trace the path leading to final formation of the Strouhal vortices. All three stress components share similar orders of magnitude with their maximums reached near the

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downstream closure limit of the formation region. However, their regional distributions distinctly differ.
Quantitatively, the present total Reynolds stress results agree with the summed periodic and random data reported in Cantwell and Coles (1983) for Re=140,000. Moreover, these results agree with the numerical data at Re=525 (Mittal and Balachandar, 1995), Re=2,600 (Prasad and Williamson, 1997), Re=3,400 (Jordan, 1996) and Re=3,900 (Beaudan and Moin, 1994) which extends similarity of the turbulent stress statistics over the sub-critical range 525≤Re≤140,000. Although the peak magnitudes of each stress at the various Reynolds numbers (listed in Table 1) share similar orders, their respective downstream locations vary. Most importantly, their peak locations correlate well with (Cp)b like the global wake properties discussed earlier (see Fig. 7).
Table 1. Peak Normal and Shear Reynolds Total Stresses within the Formation Region.
Total Stress
Reynolds Number
5251
34002
39003
56004
1400005
0.21
0.17
0.18
0.20
0.22
0.60
0.35
0.40
0.42
0.43
0.15
0.10
0.12
0.12
0.19
1Mittal and Balachandar (1995)
2Jordan (1996),
3Beaudan and Moin (1994)
4Present Simulation
5Cantwell and Coles, (1983).
Figure 7. Downstream Location (referenced to cylinder centerline) of Peak Reynolds Stresses and Base Pressure Coefficient as a Function of Reynolds Number (refer to Table 1 for the reference source of the data points).
Figure 8. Final Grid Used for the ΝACA 0018 Hydrofoil Simulation.
Figure 9. Typical Snapshot of the Vortex Shedding Near the Trailing Edge Tip.
Figure 10. Time-Averaged Vorticity Contours Near the Trailing Edge Tip.
5.2 NACA 0018 Hydrofoil
The grid structure generated for the hydrofoil section was 449×101×16 in the streamwise, transverse and spanwise directions, respectively (see Fig. 8). The inflow and outflow boundaries were set at 4 and 7 cord lengths, respectively. This outflow boundary was established through comparisons to the potential flow solution of the pressure variable along the section surface. Like the cylinder test case, the field lines were cluster towards the hydrofoil section to insure a minimum boundary condition in wall units of Δy+<4 normal to the surface. The spanwise length was set at π units with periodic end conditions.
The periodic shedding of vortex structures off the trailing edge of the NACA 0018 hydrofoil are shown in Fig. 9. The contours depict instantaneous

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vorticity with homogeneity assumed in the spanwise direction. The vortical structures rapidly traverse the trailing edge of the hydrofoil with subsequent shedding near the tip. In the immediate wake region, successive vortices closely interact such that their mutual interference causes breakdown or pairing. These observations were also detected in the flow visualization tests reported by Hayashi et al. (1995). Notably, this behavior is particularly unlike the vortex shedding process of a cylinder discussed above where the shed vortices are a several diameters apart and remain strongly coherent for many diameters downstream of formation.
The spanwise vortical structures traverse along the trailing edge and their associated shedding statistics are not as well-understood as the vortex formation process behind a circular cylinder. The time-averaged vorticity contours shown in Fig. 10 suggest a fully attached boundary layer. Only extremely close to the hydrofoil surface (over a few grid spaces) does the mean vorticity contours indicate pockets of minor circulation. Since the instantaneous separation point strongly oscillates near the maximum hydrofoil thickness, its defined time-averaged location appears absent.
Spanwise-averaged profiles of the total lift (CL) and total drag (CD) are plotted in Fig. 11 over approximately eight shedding cycles. The lift profile depicts minor fine-scale oscillations which suggests little sign of turbulent activity or even transition to turbulence along the trailing edge. While the mean drag coefficient is approximately 0.035, the net lift coefficient is 0.18. Based on the hydrofoil thickness (t), the lift profile gives a Strouhal number (St=f·t/U∞) of 0.18 which compares favorably with 0.19 as measured by Hayashi et al. (1995) and 0.21 as numerically predicted by Hodge et al. (1978) at zero angle-of-attack..
The time-averaged pressure coefficient (Cp) in Fig. 12 is compared to the computational results at Re=41,400 as reported by Thompson et al. (1976) for a fully-attached steady laminar boundary layer. The disparity between the pressure and suction sides agrees with the net lift coefficient given above by the total lift profile. The dash line in the figure depicts results obtained if the flow is assumed to be inviscid.
Discernible turbulent activity of the NACA 0018 hydrofoil at Re=35,000 is confined primarily to the trailing edge tip as illustrated in Fig. 13. Contours
Figure 11. Profiles of Total Lift and Drag Coefficient Over 7 Shedding Cycles.
Figure 12. Profiles of the Pressure Coefficient Over 7 Shedding Cycles.
of the horizontal turbulent intensities are shown in Fig. 13a while the shear components are depicted in Fig. 13b. Further downstream in the near wake region as well as along the trailing edge, some turbulent activity is indicated but their levels are extremely low. Thus, although the trailing flow has massively separated it remains essentially laminar over a majority of the trailing edge at this Re.
Figure 13. Contours of Reynolds Stresses; (a) Streamwise, (b) Shear

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6. FINAL REMARKS
The impetus of the present large-eddy simulations was to investigate the turbulent physics of the near wake behind a circular cylinder and a NACA 0018 hydrofoil. The governing LES equations and subgrid-scale (SGS) turbulence model were formulated in curvilinear coordinates by performing the transformation operation of the full-scale resolution equations prior to their filtering.
The LES investigation of the circular cylinder extends our understanding of the wake physics and provides useful analyses for engineering design. We know that the streamwise “fingered” structures and spanwise vortical structures appear to scale according to the helicity. Also, the Reynolds stress statistics share equivalent orders-of-magnitude and similar distributions over the entire range of sub-critical Reynolds numbers. Finally, the location of the peak Reynolds stresses within the formation region scale according to the magnitude of the downstream base pressure coefficient.
With the exception of the trailing edge tip region, the flow is primarily laminar at this Re. The flow separates just after the maximum thickness of the section thereby generating large-scale separation bubbles which transverse the trailing edge. Within the near wake, these large-scale structures interact and eventually pair or apparently breakdown. They do not remain coherent far downstream. Significant turbulent characteristics are confined to a region surrounding the trailing edge tip. Relative to the mean flow, peak intensities are generally under 10 percent.
ACKNOWLEDGMENTS
The author gratefully acknowledges the support of the Office of Naval Research (Dr. L.P. Purtell, Scientific Officer), Contract No. N0001497-WX20346 and the In-house Laboratory Independent Research Program (Dr. S.Dickinson, Coordinator) at the Naval Undersea Warfare Center Division Newport.
REFERENCES
1. Bays-Muchmore, B. and Ahmed, A., (1993), “On Streamwise Vortices in Turbulent Wakes of Cylinders,” Physics of Fluids, A. 5, pp. 387–392.
2. Beaudan, P. and Moin, P., (1994), “Numerical Experiments on the Flow Past a Circular Cylinder a Sub-Critical Reynolds Number,” Report No. TF-62, Stanford University, Stanford, CA.
3. Cantwell, B. and Coles, D., (1983), “An Experimental Study of Entrainment and Transport in the Turbulent Near Wake of a Circular Cylinder,” Journal of Fluid Mechanics, Vol. 136, pp. 321–374.
4. Germano, M., Piomelli, U., Moin, P., and Cabot W.H., (1991), “A Dynamic Subgrid-Scale Eddy Viscosity Model,” Physics of Fluids, A. 3, pp. 1760–1765.
5. Gerrard, J.H., “The Wakes of Cylindrical Bluff Bodies at Low Reynolds Number,” Phil. Transactions Royal Society of London, Vol. 288, pp. 351–382.
6. Hayashi, H., Kodama, Y., Fukano, T. and Ikeda, M., (1995), ‘Relation between Wake Vortex Formation and Discrete Frequency Noise,’ Separated and Complex Flows, FED-Vol. 217, pp. 107–114.
7. Hodge, J.K., Stone, A.L. and Miller, T.E., (1978), “Numerical Solution for Airfoils Near Stall in Optimized Boundary-fitted Curvilinear Coordinates,” AIAA, Vol. 17, No. 5, pp. 458–464.
8. Jordan, S.A., (1996), ‘A Large-Eddy Simulation Methodology for Incompressible Flows in Complex Domains,’ NUWC-NPT Technical Report 10, 592.
9. Jordan, S.A. and Ragab, S.A., (1996), “An Efficient Fractional-Step Technique for Unsteady Three-Dimensional Flows,” Journal of Computational Physics, Vol. 127, pp. 218–225.
10. Jordan, S.A. and Ragab, S.A., (1998), “A Large-Eddy Simulation of the Near Wake of a Circular Cylinder,” Journal of Fluids Engineering, (to appear).
11. Kato, C., Iida, A., Takano, Y. Fujita, H. and Ikegawa, M., (1993), “Numerical Prediction of Aerodynamic Noise Radiated from Low Mach Number Turbulent Wake,” AIAA 93–0145.
12. Lilly, D.K., (1992), “A Proposed Modification of the Germano Subgrid-Scale Closure Method,” Physics of Fluids, A. 4, pp. 633–635.
13. Mittal, R. and Moin, P. (1997), “Suitability of Upwind-Biased Finite Difference Schemes for Large-Eddy Simulation of Turbulent Flows”, AIAA Journal, Vol 35, No. 8, pp. 1415–1417.

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14. Mittal, R. and Balachandar, S. (1996), “Effect of Three-dimensionality on the Lift and Drag of Nominally Two-dimensional Cylinders”, Physics of Fluids, Vol 7, No. 8, pp. 1841–1865.
15. Prasad, A. and Williamson, C.H.K., (1997), “Three-Dimensional Effects in Turbulent Bluff-Body Wakes,” Journal of Fluid Mechanics, Vol. 343, pp. 235–265.
16. Smagorinsky, J., (1963), “General Circulation Experiments with the Primitive Equations, I. The Basic Experiment,” Monthy Weather Review, Vol. 91, pp. 99–164.
17. Sugavanam A. and Wu, J.C., (1980), “Numerical Study of Separated Turbulent Flow over Airfoils,” AIAA, Vol. 20, No. 4, pp. 464–470.
18. Thompson, J.F., Thames, F.C., Hodge, S., Shanks, S.P., Reddy, R.N. and Mastin, C.W., (1976), ‘Solutions of the Navier-Stokes Equations in Various Flow Regimes on Fields Containing and Number of Arbitrary Bodies Using Boundary-fitted Coordinates Systems, 5th International conference on Numerical Methods in Fluid Dynamics, Euschede, Netherlands
19. White, F.W., (1974), Viscous Fluid Flow, McGraw-Hill, NY.
20. Zhou Y. and Antonia, R.A., (1993), “A Study of Turbulent Vortices in the Near Wake of a Cylinder,” Journal of Fluid Mechanics, Vol. 253, pp. 643–661.
DISCUSSION
J.Grant
Naval Undersea Warfare Center, USA
This paper is an important contribution to the computation of complex flows. The author begins by providing a clear description of the form of the filtered equations in the context of a body-fitted coordinate system, including the formulation of the dynamic subgrid scale model. The two cases presented, flow past a right circular cylinder (Re=5600) and flow past a NACA 0018 hydrofoil (Re=35000), are among the most complex yet reported in the literature.
The computed pressure distribution for the former case compares well with measurements made at similar Reynolds numbers, indicating that the separation point was accurately depicted by the computations. The computed pressure distribution for the hydrofoil displays a similarly good agreement with relevant experimental results.
The spatial distributions shown in Figure 6 of the phase-averaged Reynolds stresses will be useful for understanding the momentum budget for the cylinder wake vortices in the formation region. The results reported in Table 1 and Figure 7 concerning the magnitude and location of maxima suggest that the spatial distribution plots are reasonably quantitatively accurate; comparison of cross-wake profiles of the stresses with observed data would strengthen this point. Although the stated focus of this paper is on the turbulent statistics, a plot of computed and measured cross-wake variation of the mean flow would be a useful supplement to the displayed results.
There is quite a bit of ongoing discussion in the literature concerning the impact of numerical error on the results of LES calculations, and the author could have used this paper as an opportunity for contribution to this subject. First, a plot in the format of Figure 4 (a) showing contours of the ratio of truncation error for the advection term to the magnitude of the viscous term would be very helpful in understanding the relative roles of the subgrid scale model and numerical viscosity, especially as the grid spacing increases away from the cylinder. (Truncation error could be estimated by computing the leading term.) Second, a plot of the spectrum of the kinetic energy showing the roll-off—or lack of roll-off—at the higher resolved wavenumbers would further comment on this issue.
Overall, this paper displays the impressive potential for subgrid scale modeling to compute complex unsteady flows at relevant Reynolds numbers.
AUTHOR’S REPLY
First of all, I would like to thank Dr. Grant for expressing his appreciation of the paper. It does indeed present useful information regarding the Reynolds stress fields downstream of a circular cylinder and a NACA 0018 hydrofoil. I am currently focused on the cross-wake statistics downstream of these shapes, as recommended by Dr. Grant, and hope to publish the respective LES results at the next Navy symposium.
As noted by Dr. Grant, the accuracy of the convective term and its instantaneous as well as statistical interaction with the SGS term is crucial to any LES computation. The associated

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metric coefficients which accompany the transformation operation should be evaluated numerically to at least fourth-order. Choice for evaluating the derivative itself, however, depends largely on treatment of the GS stress field.
For example, contributions of a Leonard-type term are negligible at the higher resolved wavenumbers if one chooses to discretize the convective term to the second-order. One can understand this fact by comparing the energy spectrum of a box-filtered field to that of the corresponding Leonard-term. In the present paper, a third-order upwind biased scheme was combined with the dynamic model which represented the turbulent scales unresolved by the grid spacing. Although previously shown by others, severe damping, or even dumping, of turbulent energy at the higher resolved scales did not occur under the present grid resolution. This observation is evident by the close comparisons attained between the LES and experimental data (see Jordan and Ragab, 1998).
To address the truncation error issue, Table 1 lists the relative contributions of the convective, diffusive, SGS, truncation error and artificial dissipative terms in the computations of the cylinder wake. These values depict time-averages taken within the turbulent wake region over approximately 8 shedding cycles. Clearly the wake dynamics are dominated by convection. For the present spatial resolution, the SGS term contributes on the order of the diffusion term. This fact is further emphasized in Figure 1, which shows the instantaneous scaled turbulent eddy viscosity along the wake centerline taken at a particular instant in time.
Table 1: Phase-averaged Terms (Re=5,600); Convection (Cv), Molecular Diffusion (Dm), Subgrid Scale Stress (SGS), Truncation Error (TE) and Artificial Dissipation (Da)
Cv
Dm
SGS
TE
Da
O(1)
O(10−1)
O(10−1)
O(10−2)
O(10–2)
Figure 1. Phase-averaged SGS Model Results; SGS/Dm through Wake Centerline.
One can conclude from these results that the SGS term does indeed participate in the LES computations of the cylinder wake flow, but generally on the order of the diffusion term. Under-prediction of the Reynolds stress statistics in the coarse-grid regions is directly attributed to the inability of the SGS model to accurately account for the turbulent scales that encompass the inertial sub-range and dissipative range of the energy spectrum.
Reference
Jordan, S.A. and Ragab, S.A. (1998), “A Large-Eddy Simulation of the Near Wake of a Circular Cylinder,” Journal of Fluids Engineering, V120, No. 2, pp. 243–252.

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Figure 3: Detailed view of the grid near the nose of the wing
Figure 4: k−ε model- Particle traces in the symmetry plane and on the flat plate (the pressure contours are visible on the flat plate)
traces in the plane of symmetry and on the flat plate are shown in Figs. 4 and 5. The Rij−ω solution is characterised by a spiral vortex which is not present in the k−ε solution. Both models predict a primary line of separation located at the intersection between the three-dimensional surface and the flat plate. However, the low-shear stress line which was observed in the experiments between the wing and the primary line of separation is only visible in the second-moment solution, indicating that the secondary motion predicted by the anisotropic turbulence closure is far more intense.
Figure 5: Rij−ω model- Particle traces in the symmetry plane and on the flat plate (the pressure contours are visible on the flat plate)
Figure 6: Measured pressure coefficient on the flat plate, (a) measurements, (b) calculated for freestream, (from [3])
4.3 Pressure distribution and wall-flow on the flat plate
Figure 6 shows contours of mean static pressure coefficient Cp measured on the flat plate in the vicinity of the nose of the wing. An interesting feature is the distortion of the measured contours that occurs close to the wing. This distortion is due to the lowering of the pressure in the junction vortex. Figure 7 shows the static pressure coefficient Cp calculated with the two different turbulence closures, namely, the Chen & Patel k−ε model and the newly proposed Rij−ω closure. Whereas the computations based

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Figure 7: Pressure coefficient on the flat plate, (a) Rij−ω model, (b) Chen & Patel k−ε model
on the isotropic closure do not reveal any distortion of the pressure coefficient, the pressure contours obtained with the Rij−ω closure are significantly distorted close to the wing. This pressure bulge is clearly related to the far more intense vortical motion predicted by the second-moment closure in the core of the horseshoe structure.
Figure 8 displays a surface oil-flow visualization performed in the vicinity of the nose of the wing. A line of separation emanates from a saddle point located at 0.47T (T is the maximum thickness of the wing) upstream of the wing leading edge. This line is less intense than a second line emanating closer to the nose of the wing from a point located at 0.28T upstream of the wing. According to Devenport & Simpson [3], this line is not a separation line but a line of low streamwise shear which divides the separated flow into two zones: a region of high streamwise shear stresses near the wing and a strip of lower shear stresses upstream.
Figure 9 shows the velocity vectors and the limiting streamlines on the wall around the wing nose provided by the two turbulence models on the same grid. The convergence of the first line of separation is more pronounced with the Rij−ω model than with the k−ε model. The most interesting feature is the dramatic difference between the two turbulence closures concerning the inner line of low streamwise shear stress. Whereas the k−ε solution does not reveal any second line of convergence, the anisotropic Rij−ω solution
Figure 8: Surface oil-flow visualization on the flat plate (from [3])
clearly captures the correct topology of the wall flow. The line of low streamwise shear stress is well marked, located at 0.21T upstream of the wing, wraps around the leading edge and merges with the primary line of separation alongside the wing. The strong variation of the shear is clearly illustrated by the velocity field close to that line.
For the sake of brevity, no comparisons on the velocity profiles and turbulent stresses in the vertical plane of symmetry are provided in the present paper. However, it is important to mention that the comparisons revealed large discrepancies between both numerical results and the experiments near the nose of the wing (−0.30<X/T<−0.1536). The intensity of normal turbulent stresses and was severely underestimated by both turbulence closures. This deficiency must be correlated with the bi-modal structure of the flow in this region.
4.4 Results on X=cst. planes
Extensive mean and fluctuating velocity measurements were also conducted by Fleming & al. [4]

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Figure 9: Wall-limiting velocity vectors on the flat plate, (a) Rij−ω model, (b) Chen & Patel k−ε model
Figure 10: Contours of U/Uref at X/C=0.64, (a) Experiments; (b) k−ε model
at several Χ/C=cst. planes adjacent to the wing and in the wake. These measurements made it possible to characterise the horseshoe vortex development. For the sake of brevity, only two stations, namely Χ/C=0.64; 1.50 are chosen to compare the respective performances of k−ε and Rij−ω models. Figures 10 and 11 show the near-wing contours of U/Uref with both turbulence models at Χ/C=0.64.
The distortion of the iso-velocity contours observed in the measurements are due to the secondary motion which transports higher momentum fluid from the edge of the boundary layer to the near-wall region. This trend is severely underestimated by the k−ε solution and accurately reproduced by the anisotropic Rij−ω model which yields a more intense vortical crossflow. Figures 12 and 13 show the contours of the rms Reynolds normal stress u′/Uref at the same sta
Figure 11: Contours of U/Uref at Χ/C=0.64, (a) Experiments; (b) Rij−ω model
Figure 12: Contours of u′/Uref at Χ/C=0.64, (a) Experiments; (b) k−ε model
tion. A local maximum is located in the region where the distorsion of U/Uref contours occurs, indicating that this local peak of turbulence is probably due to the increased mixing of the boundary layer fluid by the horseshoe vortex. The k−ε solution exhibits a large zone of high normal stress located in the corner between the wing and the flat plate. This high turbulence intensity generates an excessive level of turbulent viscosity which absorbs the secondary motion in the core of the horseshoe vortex. On the contrary, the anisotropic second-moment closures provides a much more physical solution since the local peak of u’ is accurately positionned, even if its maximum level is Somewhat underestimated.
Figures 14 and 15 show the isowakes in the near wake at Χ/C=1.50 and figures 16 to 17 display the corresponding Reynolds normal stress u’. Here, under the joint action of near-wall pressure-induced crossflow and vortex-induced secondary flow, the distortions of iso-velocity contours are less pronounced, indicating that the longitudinal vorticity is somewhat relaxed. This tendency is accurately reproduced by the k−ε solution. However, this limited success is rather due to the fact that the longitudinal vorticity field predicted by this closure at the upstream stations was severely underestimated. Contrary to

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Figure 13: Contours of u′/Uref at Χ/C=0.64, (a) Experiments; (b) Rij−ω model
Figure 14: Contours of U/Uref at Χ/C=1.50, (a) Experiments; (b) k−ε model
the isotropic model, the second-moment solution indicates a slower than measured decay of the secondary motion.
This trend was already noticed by Sotiropoulos & Patel [40] who used the original Shima model to compute the growth and decay of longitudinal vortices in a straight circular-to-rectangular transition duct. Therefore, it is plausible that this inability to capture the recovery of the flow in the wake should be attributed to the modeling of the Reynolds-stress trans
Figure 15: Contours of U/Uref at Χ/C=1.50, (a) Experiments; (b) Rij−ω model
Figure 16: Contours of u′/Uref at Χ/C=1.50, (a) Experiments; (b) k−ε model
Figure 17: Contours of u′/Uref at Χ/C=1.50, (a) Experiments; (b) Rij−ω model
port equations.
4.5 Velocity profiles and turbulent stress profiles along a specific line
In a more recent study, Ölçmen & Simpson [5] have measured the velocity profiles and all Reynolds stresses at seven stations located along a line determined by the mean velocity vector component parallel to the wall in the layer where is maximum. The locations of measurements are described in figure 18. All the velocity and Reynolds stress components are given in their respective local free-stream coordinate axes. In the computations, the free-stream coordinates are provided by the velocity directions at the first point (from the wall) where U=0.995Uext. For the sake of brevity, only three stations, namely, stations 3, 5 and 7, are selected for discussion in this paper. All the results are plotted with semi-logarithmic coordinates.
4.5.1 Mean velocity profiles
Figs. 19, 20 & 21 display the mean velocity profiles at stations 3, 5 & 7. All the figures show that both tur-

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Figure 18: Locations of the velocity and stress measurements
bulence models simulate reasonably well the general trends of the mean flow field.
However, the Rij−ω closure provides systematically a better representation of the three-dimensionality of the flow. The longitudinal U-component is always in better accordance with the experiments and the W-component is more intense than the one calculated with the k−ε model. For
Figure 19: Velocity profiles in free-stream coordinates at station 3. Solid line: Reynolds-stress model; dashed line: k−ε model; symbols: experiment
Figure 20: Velocity profiles in free-stream coordinates at station 5. Solid line: Reynolds-stress model; dashed line: k−ε model; symbols: experiment
Figure 21: Velocity profiles in free-stream coordinates at station 7. Solid line: Reynolds-stress model; dashed line: k−ε model; symbols: experiment

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Figure 22: Normal stress components in free-stream coordinates at station 3. Solid line: Reynolds-stress model; dashed line: k−ε model; symbols: experiment
instance, at station 5 (Fig. 20), the k−ε model predicts for the W-component of the velocity a minimum value of −0.15, whereas the Rij−ω closure provides a value of −0.18, the experimental value being around −0.21. Therefore, even if the Rij−ω model still overestimates the U-component and under-estimates the W-component, it is clear that the even imperfect simulation of turbulence anisotropy has dramatically increased the three-dimensionality of the flow towards the solution revealed by the measurements.
4.5.2 Normal turbulent stresses
Figs. 22, 23 & 24 show the normal turbulent stresses at the same stations 3, 5 & 7. As expected, the k−ε closure is not able to discriminate between the three normal stresses which are roughly identical to 2k/3. The normal stresses computed with the new Rij−ω closure agree reasonably well with the experimental results. From stations 3 to 7, increases continuously in the vicinity of the wall and decreases in the outer region. This trend is closely simulated by the Rij−ω closure although the magnitude of that stress is slightly underestimated.
The correlation increases more gradually and reaches a plateau from y/T comprised between 10−2 and 510−1. This particular behaviour is well simulated by the anisotropic closure which, hereagain, underestimates the magnitude of this normal stress.
Figure 23: Normal stress components in free-stream coordinates at station 5. Solid line: Reynolds-stress model; dashed line: k−ε model; symbols: experiment
Figure 24: Normal stress components in free-stream coordinates at station 7. Solid line: Reynolds-stress model; dashed line: k−ε model; symbols: experiment

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Figure 25: Shear stress components in free-stream coordinates at station 3. Solid line: Reynolds-stress model; dashed line: k−ε model; symbols: experiment
The correlation increases more rapidly near the wall and reaches a plateau closer to the wall than . This behaviour is well captured by the anisotropic model even if the computed normal stress is slightly underevaluated in the outer region of the boundary layer.
4.5.3 Turbulent shear stresses
Figs. 25, 26 & 27 show the turbulent shear stresses at the same stations as before. As expected, both models provide similar simulations of and shear stresses.
The slight irregularities which can be observed on the k−ε results are due to the lack of continuity of derivatives through the boundaries between the two layers defined in Chen & Patel model. is accurately predicted by both models but is severely overerestimated in the outer region of the boundary layer. The most remarkable feature in these comparisons is the dramatic difference concerning the computations of has a local maximum value close to the wall and starts developping to reach an adimensional maximum value of 0.003 at station 5 with a change of sign occurring at Y/T≈210−2 for most of the stations. This behaviour is absolutely ignored by the computations based on the k−ε closure which even do not predict the negative value of in the outer region at station 7 (Fig. 27).
Figure 26: Shear stress components in free-stream coordinates at station 5. Solid line: Reynolds-stress model; dashed line: k−ε model; symbols: experiment
Figure 27: Shear stress components in free-stream coordinates at station 7. Solid line: Reynolds-stress model; dashed line: k−ε model; symbols: experiment

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On the other hand, the computations based on the anisotropic Rij−ω closure represent fairly well this evolution, even if the maximum value of this turbulent shear stress is hereagain slightly overestimated. The simulation of turbulence anisotropy creates the mechanism for predicting the correct near-wall behaviour of the turbulent shear stress that is the most correlated to the three-dimensionality of the flow. The complex coupling between the normal and shear turbulent stresses is correctly accounted for by the present Rij−ω closure which provides a dramatically new physical solution much closer to the measurements than the solution obtained with k−ε model.
5 Summary and conclusion
The two-equation, two-layer, k−ε model of Chen & Patel and a new seven-equation Reynolds-stress transport model based on the model of Shima [18] in which a new transport equation for the turbulent frequency ω was implemented, were employed to calculate the flow around a wing-body junction. For both models, the flow was resolved up to the wall and all numerical characteristics (spatial discretization, numerical schemes, initial and boundary conditions) were held invariant to evaluate the importance of turbulence model in the computation of a wing-body junction flow. Each computations were compared with the extensive experimental database generated by the Virginia Polytechnic Institute during more than eight years. This systematic evaluation lead to the following conclusions:
All the comparisons with global and local quanti-| ties, ranging from the velocity components to the Reynolds-stress tensor components have established the clear superiority of the second-moment closure. This new Reynolds-stress transport model accurately reproduced the anisotropic behaviour of the normal Reynolds stress components, which led to a clear amplification of the longitudinal vorticity. Moreover, the Reynolds stress anisotropy was also responsible for the strong augmentation of the shear stress close to the horizontal plane. This result clearly demonstrated that second-moment calculations were particularly well suited to three-dimensional vortical flows involving several predominant flow gradients.
As expected, near the leading edge of the wing, the statistical second-moment model provided a solution which was significantly different from what was observed by [3]. The simulation of the flow in that region is somewhat out of the scope of a steady second-moment turbulent simulation since the flow appears to be characterised by a large scale unsteadiness. It would be of great interest to use this database to assess the actual potentialities of Coherent-Structure-Capturing methods [41] to determine the respective domain of applications and computational resources required by statistical second-moment closures and Large-Eddy-Simulations methods.
6 Acknowledgments
Tabulated experimental data and flow visualization photographs were kindly provided by Prof. R.L. Simpson. Thanks are due to the Scientific Committee of IDRIS and the DS/SPI for attributions of Cpu on the VPP and T3E on which most of the calculations were performed.
References
[1] I.M.M.A Shabaka and P.Bradshaw, “Turbulent flow measurements in an idealized wing/body junction,” ΑΙΑA J., Vol. 19, No. 2, 1981, pp. 131–132.
[2] S.C.Dickinson, “An experimental investigation of appendage-flat plate junction flow,” DTNSRDC Report 86/052, Vol. 1–2, 1986.
[3] W.J.Devenport and R.L.Simpson, “Time-dependent and time-averaged turbulence structure near the nose of a wing-body junction,” Journal of Fluid Mechanics, Vol. 210, 1990, pp. 23–55.
[4] R.L.Simpson J.L.Fleming and W.J.Devenport, “An experimental study of a turbulent wing-body junction and wake flow,” Experiments in Fluids, Vol. 14, 1993, pp. 366–378.
[5] S.M.Ölçmen and R.L.Simpson, “An experimental study of a three-dimensional pressure-driven turbulent boundary layer,” Journal of Fluid Mechanics, Vol. 290, 1995, pp. 225–262.
[6] W.R.Briley and H.McDonald, “Computations of three-dimensional horseshoe vortex flow using the Navier-Stokes equations,” Proceedings of the 7th ICNMFD Conf. Stanford, 1980.

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[7] T.R.Govindan J.J.Gorski and B.Lakshminarayana, “Comparison of three-dimensional turbulent shear flow in corners,” ΑΙΑA J., Vol. 23, No. 12, 1985, pp. 685–692.
[8] R.W.Burke, “Computation of turbulent incompressible wing-body junction flow.” AIAA Paper 89–0279, 1989.
[9] C.H.Sung and C.I.Yang, “Validation of turbulent horseshoe vortex flows,” Proceedings of the 17th ONR Symposium on Naval Hydrodynamics, The Hague, The Netherlands, 1988, pp. 241–255.
[10] H.C.Chen and V.C.Patel, “The flow around wing-body junctions,” Proceedings of the 4th Symposium on Numerical and Physical Aspects of Aerodynamic Flows, Long Beach, CA, 1989, pp. 16–19.
[11] G.B.Deng and J.Piquet, “Computation of the flow past an airfoil-flat plate junction,” Int. J. Num. Meth. Fluids, Vol. 15, 1992, pp. 99–124.
[12] H.C.Chen, “Assessment of a Reynolds stress closure model for appendage-hull junction flows,” Journal of Fluids Engineering, Vol. 117, 1995, pp. 557–563.
[13] N.Shima, “A Reynolds-stress model for near-wall and low-Reynolds-number regions,” Journal of Fluids Engineering, Vol. 110, 1988, pp. 38–44.
[14] C.G.Speziale, S.Sarkar, and T.B.Gatski, “Modeling the pressure-strain correlation of turbulence: An invariant dynamical systems approach,” Journal of Fluid Mechanics, Vol. 227, 1991, pp. 245–272.
[15] T.Rung S.Fu, Z.Zhai and F.Thiele, “Numerical study of flow past a wing-body junction with a realizable nonlinear EVM,” Proceedings of the 11th Turbulent Shear Flows, 1997, pp. 6–7–6–12.
[16] C.G.Speziale, “On turbulent secondary flows in pipes of non circular sections,” Int. J. Eng. Sci., Vol. 20, 1982, pp. 863–872.
[17] B.E.Launder and N.Shima, “Second-Moment Closure for the Near-Wall Sublayer: Development and Application,” AIAA Journal, Vol. 27, 1989, pp. 1319–1325.
[18] N.Shima, “Prediction of turbulent boundary layers with a second moment closure,” Journal of Fluids Engineering, Vol. 115, 1993, pp. 1–27.
[19] B.E.Launder and D.P.Tselepidakis, “Progress and Paradoxes in Modelling Near-Wall Turbulence,” Proceedings of the 8th Symposium on Turbulent Shear Flows, Munich, 1991, pp. 29–1–29–6.
[20] T.J.Craft and B.E.Launder, “Improvments in near-wall Reynolds stress modelling for complex flow geometries,” Proceedings of the 10th Turbulent Shear Flows, 1995, pp. 20–25–20–30.
[21] S.Jakirlić and K.Hanjalić, “A Second-Moment Closure for Non-Equilibrium and Separating High- and Low-Re-Number Flow,” Proceedings of the 10th Symposium on Turbulent Shear Flows, Pennsylvania, 1995, pp. 23–25–23–30.
[22] J.Kim, P.Moin, and R.Moser, “Turbulence Statistics in Fully Developed Channel Flow at Low Reynolds Number,” J. Fluid Mech., Vol. 177, 1987, pp. 133–166.
[23] G.B.Deng and M.Visonneau, “Near-wall modelization for dissipation in second-moment closures,” Proceedings of the 11th Turbulent Shear Flows, 1997, pp. P2–101–P2–106.
[24] G.B.Deng and M.Visonneau, “Assessment of second-moment turbulence closures for three-dimensional vortical flows,” Proceedings of the 1997 ASME Fluids Engineering Division Summer Meeting, 1997.
[25] K.Wieghardt and J.Kux, “Nomineller nachstrom auf grund von windkanal versuchen,” Jahrb. der Schiffbau Technischen Gesellschaft (STG), 1980.
[26] K.Wieghardt, “Kinematics of ship wake flow,” Proceedings of the 7th David Taylor Memorial Lecture, DTNSRDC Report 81/093, 1982.
[27] W.J.Kim and V.C.Patel, “Origin and decay of longitudinal vortices in developing flow in a curved rectangular duct,” Journal of Fluids Engineering, Vol. 116, 1994, pp. 45–52.
[28] Computation of Turbulent Boundary Layers—1968 AFOSR-IFP-Stanford Conference, D.E. Coles & E.A.Hirt, ed., 1968.

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[29] B.S.Baldwin and T.J.Barth, “A one equation turbulence transport model for high Reynolds number wall-bounded flows,” AIAA 29th Aerospace Sciences Meeting, AIAA Paper 91–0610, 1991.
[30] H.C.Chen and V.C.Patel, “Practical near-wall turbulence models for complex flows including separation.” AIAA-87–1300, 1987.
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DISCUSSION
C.H.Sung and M.J.Griffin
Naval Surface Warfare Center, Carderock Division, USA
There is no doubt that of all the turbulence models, the Reynolds Stress model contains the most physics and should be pursued.
To avoid hot-wiring in a production code, it is important that normal wall distances do not appear in the model. We would like to hear authors’ comments on the robustness of the damping functions used. In particular, if they work in flat-plate, do they also work in body at incidence without changing their values? The damping function fw which contains normal wall distance is of particular interest. What is the prospect of eliminating the normal wall distance from this function?
We pursue nonlinear two-equation models. The vortex in front of the wing computed using Speziale’s quadratic non-linear k-w model is shown below:

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It can be seen that the predicted vortex formation compares favorably with the experiment when three different turbulence models are used, as presented by A. Rizzi & J.Vos (AIAA Journal, Vol. 36, May 1998) and shown below:
AUTHORS’ REPLY
Wall damping function is usually constructed by using one of the three parameters: y+=yuτ/ν, and Rτ=k2/(νε). In the authors’ opinion, it is not desirable to use a wall related parameter which changes during temporal or nonlinear iteration, such as uτ. Compared with Rτ, Ry is a more representative wall-distance indicator. Models based on Ry are usually more stable that those based on Rτ. Although the wall normal distance is difficult to determine exactly for complex geometries, a numerical approximation is easy to implement. The authors use a body-fitted multi-block solver that is capable of computing the wall normal distance for any given grids. More complex geometries such as the HSVA tanker have been treated without changing anything with the same code. The authors do not feel the need to replace Ry in the wall damping function for the application in complex geometries. However, when using a moving grid, the computation of wall-normal distance may become very time consuming. In this case, models based on Rτ will become attractive.
One of the most important contributions of the Reynolds stress transport model to the prediction of the wing-body junction flow is a better description of normal stress anisotropy and the exact description of the convection effect of Reynolds stress which is found to be very important in front of the wing. Results shown by the discussers are very interesting. It indicates that by taking into account the normal stress anisotropy with a nonlinear two-equation model, one can improve prediction of the horse-shoe vortex. However, the nonlinear two-equation models which are deduced from a Reynolds stress transport model by using local equilibrium assumption, are obviously unable to account for the convection effect. Consequently, other important features such as the distinction of high shear stress and low shear stress regions, the position of the pigment accumulation line, etc., captured by the Reynolds stress transport model, are unable to be predicted with nonlinear models.
Predictions with a Reynolds stress transport model given by A.Rizzi are not representative. They are probably obtained with a wall function approach, which suggests that wall function approach should not be used for complex three-dimensional flow prediction. In the present studies, the authors use a low Reynolds number model where wall reflection terms need to be added. Unfortunately, just like the wall function approach, the wall reflection terms are calibrated with simple boundary layer flow. The present study shows that they are not valid for separated flows. The shape of the horse-shoe vortex predicted by the Reynolds stress transport model is presented in the following figure. It is less satisfactory than the result obtained by the discussers with a nonlinear two-equation model, which indicates that the improvement provided by an exact description of convection effect is smaller than the deterioration introduced by the wall reflection terms.
Velocity vectors in the vertical symmetry plane.