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FIG. 15. (u'w’) shear stress profiles in the three-dimensional boundary layer flow. (a) ru,,,/5=0.075; (b) tu,,,/5=0.3; (c) tu,,,/5=0.6. — Dy- namic eddy-viscosity model; ——— Lagrangian eddy-viscosity model, - dynamic mixed model; — - — one-coefficient Lagrangian mixed model; - - - - two-coefficient Lagrangian mixed model; A DNS (Ref. 27).

Figure 15 (u'w’) shear stress profiles in the three-dimensional boundary layer flow. (a) ru,,,/5=0.075; (b) tu,,,/5=0.3; (c) tu,,,/5=0.6. — Dy- namic eddy-viscosity model; ——— Lagrangian eddy-viscosity model, - dynamic mixed model; — - — one-coefficient Lagrangian mixed model; - - - - two-coefficient Lagrangian mixed model; A DNS (Ref. 27).

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(where p is the pressure, v the kinematic viscosity, and u,, uy, and u3, or u, uv, and w, are the streamwise, wall-normal and spanwise components of velocity, respectively), which govern the dynamics of the large, energy-carrying scales of motion. The effect of the small scales upon the resolved part of turbulence appears in the subgrid-scale (SGS) stress term, where G is the filter function of characteristic width A, and the integral is extended to the entire domain D. The applica- tion of the filtering operation to the continuity and the Navier-Stokes equations gives the resolved Navier-Stokes equations, which, in incompressible flow, are
(where p is the pressure, v the kinematic viscosity, and u,, uy, and u3, or u, uv, and w, are the streamwise, wall-normal and spanwise components of velocity, respectively), which govern the dynamics of the large, energy-carrying scales of motion. The effect of the small scales upon the resolved part of turbulence appears in the subgrid-scale (SGS) stress term, where G is the filter function of characteristic width A, and the integral is extended to the entire domain D. The applica- tion of the filtering operation to the continuity and the Navier-Stokes equations gives the resolved Navier-Stokes equations, which, in incompressible flow, are
TABLE I. Simulation parameters. Two cases were examined: A two-dimensionai plane channel flow, at Re,= du,/v=180 and 1 050, and a three- dimensional (3D) boundary layer. In all cases, periodic boundary conditions were applied in the streamwise and spanwise directions, while the no-slip conditions were used at the solid walls. The three-dimensional boundary layer is a temporally developing flow obtained by imposing an impul- sive spanwise motion, with magnitude equal to 47% of the initial mean centerline velocity, to the wall of a fully devel- oped plane channel flow; the initial condition was a calcula- tion of fully developed plane channel flow at Reynolds num- ber Re,=180. These parameters matched the DNS calculations of Coleman eg al.?’ The flow was allowed to develop until a collateral state (one in which the direction of the mean velocity is the same at each y) was reached. While in the direct numerical simulation (DNS) of Coleman et al?’ only one wall was set in motion, in the present case both walls were moved in opposite directions to double the statis- tical sample. It was verified that the important statistics were not affected up to a dimensionless time tu, ,/5= 1.2 (where u,, is the value of the friction velocity before the walls are set into motion). The two configurations examined are sketched in Fig. 1. Scale-similar models were introduced by Bardina et al.,'© who assumed that the component of the SGS most active in the energy transfer from large to small scales can be estimated with sufficient accuracy from the smallest resolved
TABLE I. Simulation parameters. Two cases were examined: A two-dimensionai plane channel flow, at Re,= du,/v=180 and 1 050, and a three- dimensional (3D) boundary layer. In all cases, periodic boundary conditions were applied in the streamwise and spanwise directions, while the no-slip conditions were used at the solid walls. The three-dimensional boundary layer is a temporally developing flow obtained by imposing an impul- sive spanwise motion, with magnitude equal to 47% of the initial mean centerline velocity, to the wall of a fully devel- oped plane channel flow; the initial condition was a calcula- tion of fully developed plane channel flow at Reynolds num- ber Re,=180. These parameters matched the DNS calculations of Coleman eg al.?’ The flow was allowed to develop until a collateral state (one in which the direction of the mean velocity is the same at each y) was reached. While in the direct numerical simulation (DNS) of Coleman et al?’ only one wall was set in motion, in the present case both walls were moved in opposite directions to double the statis- tical sample. It was verified that the important statistics were not affected up to a dimensionless time tu, ,/5= 1.2 (where u,, is the value of the friction velocity before the walls are set into motion). The two configurations examined are sketched in Fig. 1. Scale-similar models were introduced by Bardina et al.,'© who assumed that the component of the SGS most active in the energy transfer from large to small scales can be estimated with sufficient accuracy from the smallest resolved
FIG. |. Sketch of the physical configurations: (a) Plane channel; (b) three- dimensional boundary layer.
FIG. |. Sketch of the physical configurations: (a) Plane channel; (b) three- dimensional boundary layer.
TABLE II. Wall shear stress.
TABLE II. Wall shear stress.
FIG. 2. Mean velocity profiles in wall coordinates in the two-dimensional plane channel, Re,=1050. Dynamic eddy-viscosity model (plane- averaged). —~ Fourier cutoff filter; --- top-hat filter, A,/8,= V6; — . — top-hat filter. A./A,=2: — logarithmic law.
FIG. 2. Mean velocity profiles in wall coordinates in the two-dimensional plane channel, Re,=1050. Dynamic eddy-viscosity model (plane- averaged). —~ Fourier cutoff filter; --- top-hat filter, A,/8,= V6; — . — top-hat filter. A./A,=2: — logarithmic law.
FIG. 3. Mean velocity profiles in wall coordinates in the two-dimensional plane channel. (a) Re,= 180, (b) Re,= 1 050.x DNS data; —- dynamic eddy-viscosity model, --- dynamic mixed model; — - — one- coefficient Lagrangian mixed model; ---- two-coefficient Lagrangian mixed model; ——— logarithmic law.
FIG. 3. Mean velocity profiles in wall coordinates in the two-dimensional plane channel. (a) Re,= 180, (b) Re,= 1 050.x DNS data; —- dynamic eddy-viscosity model, --- dynamic mixed model; — - — one- coefficient Lagrangian mixed model; ---- two-coefficient Lagrangian mixed model; ——— logarithmic law.
FIG. 4. Turbulence intensity profiles in wall coordinates in the two- dimensional plane channel. (a) Re,=180, (b) Re,=1 050. —- Dynamic eddy-viscosity model; --- dynamic mixed model; — + — one- coefficient Lagrangian mixed model; ---- two-coefficient Lagrangian mixed model; XDNS data (Ref. 36); A experimental data (Ref. 37).
FIG. 4. Turbulence intensity profiles in wall coordinates in the two- dimensional plane channel. (a) Re,=180, (b) Re,=1 050. —- Dynamic eddy-viscosity model; --- dynamic mixed model; — + — one- coefficient Lagrangian mixed model; ---- two-coefficient Lagrangian mixed model; XDNS data (Ref. 36); A experimental data (Ref. 37).
FIG. 5. Subgrid-scale shear stress in the two-dimensional plane channel. (a) Re,= 180, (b) Re,=1 050. — Dynamic eddy-viscosity model, --- dy- namic mixed model, — + — one-coefficient Lagrangian mixed model, . ..- Lagrangian two-coefficients mixed model. Lines: eddy viscosity com- ponent; lines and symbols: eddy viscosity and scale-similar components.
FIG. 5. Subgrid-scale shear stress in the two-dimensional plane channel. (a) Re,= 180, (b) Re,=1 050. — Dynamic eddy-viscosity model, --- dy- namic mixed model, — + — one-coefficient Lagrangian mixed model, . ..- Lagrangian two-coefficients mixed model. Lines: eddy viscosity com- ponent; lines and symbols: eddy viscosity and scale-similar components.
aC. 6. Subgrid-scale dissipation in the two-dimensional plane channel, =180. (a) Dissipation by the eddy-viscosity part of the SGS stresses, oe (b) dissipation by the scale-similar part of the SGS stresses, e¢2, (C) total SGS dissipation, €,,,. —- Dynamic eddy- viscosity model; --- dy- namic mixed model; — - — one-coefficient Lagrangian mixed model: - two-coefficient Lagrangian mixed model.
aC. 6. Subgrid-scale dissipation in the two-dimensional plane channel, =180. (a) Dissipation by the eddy-viscosity part of the SGS stresses, oe (b) dissipation by the scale-similar part of the SGS stresses, e¢2, (C) total SGS dissipation, €,,,. —- Dynamic eddy- viscosity model; --- dy- namic mixed model; — - — one-coefficient Lagrangian mixed model: - two-coefficient Lagrangian mixed model.
FIG. 7. Subgrid-scale dissipation in the two-dimensional plane channel, Re, = 1 050. (a) Dissipation by the eddy-viscosity part of the SGS sees €sgs5 (D) dissipation by the scale-similar part of the SGS stresses, €55,; (c) total SGS dissipation, €,,,. —- Dynamic eddy- viscosity model; --- dy- namic mixed model, — - — one-coefficient Lagrangian mixed model, - two-coefficient Lagrangian mixed model.
FIG. 7. Subgrid-scale dissipation in the two-dimensional plane channel, Re, = 1 050. (a) Dissipation by the eddy-viscosity part of the SGS sees €sgs5 (D) dissipation by the scale-similar part of the SGS stresses, €55,; (c) total SGS dissipation, €,,,. —- Dynamic eddy- viscosity model; --- dy- namic mixed model, — - — one-coefficient Lagrangian mixed model, - two-coefficient Lagrangian mixed model.
FIG. 8. Eddy-viscosity and scale-similar model coefficients in the two- dimensional plane channel. Two-coefficient Lagrangian mixed model. — Re,=1 050; --- Re,= 180. (a) C,,; (b) Coy.
FIG. 8. Eddy-viscosity and scale-similar model coefficients in the two- dimensional plane channel. Two-coefficient Lagrangian mixed model. — Re,=1 050; --- Re,= 180. (a) C,,; (b) Coy.
FIG. 10. Time history of the wall shear stress and average turbulent kinetic energy in the three-dimensional boundary layer flow. All quantities are ini- tialized by their initial values. (a) +, /7,.,; (0) K/K,. — Coarse DNS; --- Smagorinsky model; — - — dynamic eddy-viscosity model; - - - - dynamic mixed model; - - - - - one-coefficient Lagrangian mixed model; --- two-coefficient Lagrangian mixed model; A DNS (Ref. 27).
FIG. 10. Time history of the wall shear stress and average turbulent kinetic energy in the three-dimensional boundary layer flow. All quantities are ini- tialized by their initial values. (a) +, /7,.,; (0) K/K,. — Coarse DNS; --- Smagorinsky model; — - — dynamic eddy-viscosity model; - - - - dynamic mixed model; - - - - - one-coefficient Lagrangian mixed model; --- two-coefficient Lagrangian mixed model; A DNS (Ref. 27).
FIG. 9, Eddy-viscosity model coefficient in the two-dimensional plane channel, Re,= 1050. —- Dynamic eddy-viscosity model; --- dynamic mixed model; — - — one-coefficient Lagrangian mixed model; - - - - two-coefficient Lagrangian mixed model. (a) average; (b) root-mean-square; {c) maximum.
FIG. 9, Eddy-viscosity model coefficient in the two-dimensional plane channel, Re,= 1050. —- Dynamic eddy-viscosity model; --- dynamic mixed model; — - — one-coefficient Lagrangian mixed model; - - - - two-coefficient Lagrangian mixed model. (a) average; (b) root-mean-square; {c) maximum.
FIG. 11. Total (viscous +SGS) dissipation profiles. — Smagorinsky model, --- dynamic eddy-viscosity model, — - — one-coefficient Lagrangian model. (a) t/u,,,<0; (b) t/u,,,=0.08; (c) t/u,,=0.16.
FIG. 11. Total (viscous +SGS) dissipation profiles. — Smagorinsky model, --- dynamic eddy-viscosity model, — - — one-coefficient Lagrangian model. (a) t/u,,,<0; (b) t/u,,,=0.08; (c) t/u,,=0.16.
FIG. 13. Mean velocity profiles in the three-dimensional boundary layer flow. (a) tu,,, /8=0.075; (b) tu,,,/6=0.3; (c) tu,,,/5=0.6. — Dynamic eddy-viscosity model; --- dynamic mixed model, — - — _ one- coefficient Lagrangian mixed model; -- ++ two-coefficient Lagrangian mixed model; A DNS (Ref. 27).
FIG. 13. Mean velocity profiles in the three-dimensional boundary layer flow. (a) tu,,, /8=0.075; (b) tu,,,/6=0.3; (c) tu,,,/5=0.6. — Dynamic eddy-viscosity model; --- dynamic mixed model, — - — _ one- coefficient Lagrangian mixed model; -- ++ two-coefficient Lagrangian mixed model; A DNS (Ref. 27).
FIG. 12. Time-development of the coefficient C,,, in the three-dimensional boundary layer flow. —- y7=6; --- yf 13; - ++ - yj =57. (a) Dynamic eddy-viscosity model; (b) one-coefficient Lagrangian mixed model.
FIG. 12. Time-development of the coefficient C,,, in the three-dimensional boundary layer flow. —- y7=6; --- yf 13; - ++ - yj =57. (a) Dynamic eddy-viscosity model; (b) one-coefficient Lagrangian mixed model.
FIG. 14. Turbulent kinetic energy profiles in the three-dimensional bound- ary layer flow. (a) tu, ,/0= 0.075; (b) tu,,,/d=0.3; (c) tu,,/5=0.6. — Dynamic eddy-viscosity model, --- dynamic mixed model} — - — one-coefficient Lagrangian mixed model; - - - - two-coefficient Lagrangian mixed model; A DNS (Ref. 27).
FIG. 14. Turbulent kinetic energy profiles in the three-dimensional bound- ary layer flow. (a) tu, ,/0= 0.075; (b) tu,,,/d=0.3; (c) tu,,/5=0.6. — Dynamic eddy-viscosity model, --- dynamic mixed model} — - — one-coefficient Lagrangian mixed model; - - - - two-coefficient Lagrangian mixed model; A DNS (Ref. 27).
TABLE III. CPU time required by various models, normalized by that used by the Smagorinsky model. One important consideration regards the increased re- sources required by more complex LES models. The dy- namic model requires more operations and memory than the Smagorinsky one, since the various terms used in the con- tractions must be computed and stored. Mixed models and the Lagrangian averaging procedure require additional effort. Table III reports the CPU time (normalized by the Smagor- insky model) of each model. The numbers reported show, of course, only qualitative trends, since they are strongly implementation- and machine-dependent. In this particular case, all the computations were run on a Cray C90, and the model coefficient (or coefficients) were evaluated once per step, while both the eddy-viscosity and the scale-similar models were computed at each Runge-Kutta stage. It is quite readily seen that the additional cost of the Lagrangian models is significant.
TABLE III. CPU time required by various models, normalized by that used by the Smagorinsky model. One important consideration regards the increased re- sources required by more complex LES models. The dy- namic model requires more operations and memory than the Smagorinsky one, since the various terms used in the con- tractions must be computed and stored. Mixed models and the Lagrangian averaging procedure require additional effort. Table III reports the CPU time (normalized by the Smagor- insky model) of each model. The numbers reported show, of course, only qualitative trends, since they are strongly implementation- and machine-dependent. In this particular case, all the computations were run on a Cray C90, and the model coefficient (or coefficients) were evaluated once per step, while both the eddy-viscosity and the scale-similar models were computed at each Runge-Kutta stage. It is quite readily seen that the additional cost of the Lagrangian models is significant.
Related topics:
EngineeringPhysicsMechanicsLarge Eddy SimulationStatistical PhysicsChannel FlowPhysics of Fluids
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