A DES Procedure Applied to a Wall-Mounted Hump

This paper describes a detached-eddy simulation(DES) for the ﬂow over a wall mounted hump using MBFLO3 1,2 . The Reynolds number based on the hump chord is Re c = 9 . 36 × 10 5 with an inlet Mach number of 0 . 1 . Solutions of the three-dimensional Reynolds-averaged NavierStokes(RANS) procedure are obtained using the Wilcox k − ω equations. The DES results are obtained using the model presented by Bush and Mani 3 and are compared with RANS solutions and experimental data from NASA’s 2004 Computational Fluid Dynamics Validation on Synthetic Jets and Turbulent Separation Control Workshop 4,5 . The DES procedure exhibited a three-dimensional ﬂow structure in the wake, with a 13 . 65% shorter mean separation region compared to RANS and a mean reattachment length that is in good agreement with experimental measurements. DES predictions of the pressure coeﬃcient in the separation region also exhibit good agreement with experiment and are more accurate than RANS predictions.

total energy e internal energy H total enthalpy h static enthalpy k turbulent kinetic energy p pressure P r Prandtl number P r t turbulent Prandtl number S ij mean strain-rate tensor u i velocity component V velocity magnitude µ coefficient of viscosity µ T turbulent coefficient of viscosity ω turbulent dissipation frequencŷ τ ij total shear stress tensor τ ij laminar shear stress tensor τ R ij Reynolds shear stress tensor ρ density T u Freestream turbulence level as % l dis dissipation length scale C des DES coefficients I. Introduction S imulation of "steady" and "unsteady" flow of aerodynamic bodies has matured a great deal over the past decade. Aerodynamic performance and flow structures can be predicted with acceptable accuracy except

II. Governing Equations
The unsteady, Favre-averaged governing flow-field equations for an ideal, compressible gas in the righthanded, Cartesian coordinate system using primary variables are used in the MBFLO code. The threedimensional continuity, momentum, and energy equations can be written in conservative form as follows, S ij = 1 2 Since we are dealing with compressible flows, we require an equation of state to relate the energy with pressure and enthalpy.
Additional governing equations as developed by  are used for the transport of turbulent kinetic energy and turbulence dissipation rate in regions of the flow where the computational grid or global timestep size cannot resolve the turbulent eddies. In regions of the flow where the larger-scale eddies can be resolved with the computational grid, techniques borrowed from large-eddy simulation are used to represent the viscous shear and turbulent viscosity. The large-eddy sub-grid model described by Smagorinsky 12 is modified according to the detached-eddy considerations described by Strelets 13 and Bush and Mani. 3

III. Numerical Techniques
The conservation of mass, momentum, and energy equations are solved using a Lax-Wendroff controlvolume, time-marching scheme as developed by Ni, 14 Dannenhoffer, 15 and Davis. 16,17 Numerical solutions of unsteady flows can be performed with either the explicit 14 or a dual time-step procedure 18 . These techniques are second-order accurate in time and space. A multiple-grid convergence acceleration scheme 14 is used for steady, Reynolds-averaged Navier-Stokes solutions and the inner convergence loop of unsteady simulations using the dual time-step scheme. The approach is called MBFLO and has two-dimensional, 2 axi-symmetric 19 (with and without swirl), and three-dimensional 1 versions. The three-dimensional procedure, MBFLO3P (three-dimensional, parallel), and results for a DES using the Bush and Mani algorithm for flow over a wall-mounted hump are described here. The combined second-and fourth-difference dissipation model of Jameson 20 is used in the current procedure for both the mean flow and turbulence equations. The fourth-difference dissipation is scaled by the inverse of the absolute value of the mean strain rate squared. This function decays the numerical dissipation in all viscous flow regions, including boundary layers, wakes, large eddies, secondary flows, etc.
In the MBFLO suite of codes parallelization is performed using the Message Passing Interface (MPI) library. 21 Figure 1 and Table 1 show the typical speed-up and associated efficiencies as functions of the number of processors for the MBFLO3P code. The configuration used to generate this data was similar to that shown below in the results section where the computational grid consisted of 1.80 million grid points. The data was generated on a Linux cluster consisting of 3.6GHz Intel Xeon processors. Figure 1 shows that a speed-up factor of 18.06 is realized with 20 processes yielding a 90% parallel efficiency. In Table 1, we can see that efficiencies of 90% and higher can be obtained if no less then 100, 000 grid points per process are used.

IV. Results
The MBLFO suite of codes have been verified with analytical data for a series of standard test cases such as steady inviscid flow over circular bumps, turbulent flow over airfoils, laminar and turbulent flow over a flat plate, as well as axi-symmetric flows 1,2,19 . The focus of the current investigation is to demonstrate and validate the DES model for a separated flow and to determine what advantage may exist, in terms of accuracy, for DES compared to three-dimensional URANS and it's applicability in the design process.
The simulation of a turbulent flat plate, constructed in such a way as to develop the inflow boundary conditions for the wall-mounted hump, was initially performed. The length of the flat plate was determined through preliminary CFD tests so the predicted boundary layer thickness was essentially that of the experiment. The flow conditions were also set to match the wall-mounted hump with M a = 0.1 and Re c = 9.36 × 10 5 .
For the flat plate mesh, the computational domain extended from −2.80 ≤ x/c ≤ 4.25 in the streamwise direction and from 0.0 ≤ y/c ≤ 0.909 in the normal direction. There were 121 points used in the streamwise direction with a maximum stretching ratio of ∆x = 1.2. The wall spacing normal to the surface was set to 3.6 × 10 −6 and corresponds to ∆y + = 0.25. A maximum stretching ratio of 1.2 was achieved with 177 points clustered near the flat plate surface.
The simulation was performed by initializing the inlet with a uniform flow field where the non-dimensional velocity components were set to u = 1, v = 0, and w = 0 and non-dimensional density to ρ = 1. In the area over the flat plate, the velocity profile was initialized using the log-law profile, and density and energy were recomputed to keep the pressure at the freestream. During the computation, the inflow boundary condition held total pressure and total temperature constant while at the exit, static pressure was held constant. Along the flat plate surface from the inlet to the leading edge of the flat plate, an inviscid wall boundary condition was imposed with an adiabatic no-slip wall over the entire flat plate surface. The upper wall of the flow domain was also set to an inviscid wall.  To verify the predicted solution, the velocity profile was plotted in terms of the inner layer velocity and length variables u + and y + , respectively, and compared to the empirical viscous sublayer, log-law, and 1/7 th power law relations as shown in Fig. 2(a). Figure 2(b) shows the predicted velocity and experimentally measured profiles at x/c = 4.25, which corresponds to the inlet of the wall-mounted hump.
Once the inlet conditions were generated using the flat plate, the wall-mounted hump test case for turbulent flow was simulated and compared to experimental data from the NASA workshop 22 that focused on synthetic jets and turbulent separation control. The hump geometry was constructed to simulate a 20% thick Glauert-Goldschmied airfoil with a chord length of 0.472m (1.38ft.), a maximum height of 0.054m (0.176ft.), and a span of 0.584m (1.92ft.). The experimental data was obtained for M a ∞ = 0.10 and Re c = 9.36x10 5 based on the chord. The test case that was investigated in MBFLO3 was without flow control; where there was no blowing or suction in the slot. The numerical results obtain by MBLFO used a three-dimensional structured grid smoothed over the slot and with the top wall shape adjusted to approximately account for side plate blockage, as recommended by the workshop 22 . The grid shown in Fig.  3 extended from x/c = −9.20 to 3.5 in the stream-wise direction, with the hump located from x/c = 0.0 to 1.0, and from y/c = 0.0 to 0.9 in the normal direction. The mesh contained 155, 937 grid points with 881 in the stream-wise direction and 177 normal to the wall. An initial RANS simulation was performed using three planes in the span-wise direction for a grid that consisted of 467, 811 points. This allowed a nominally two-dimensional flow simulation to be performed using MBFLO3 and compared to the experiment. For true three-dimensional simulations, the original 881 × 177 grid described was used and extruded in the span-wise direction 0.15 chord lengths and can be seen in Fig. 4. The span-wise direction was meshed using 49 points with uniform spacing and corresponds to roughly two boundary layer thicknesses based on the inlet. With the aid of the turbulent flat plate simulation, the inflow velocity and density profiles for the wall-mounted hump case were generated and used to initialize the flow domain over the wall-mounted hump and helped decrease the computational time required to obtain a solution. The top of the flow domain was set with an inviscid wall boundary condition with an adiabatic no-slip wall along the south boundary starting at the inlet and leading over the hump section. Static pressure was once again held constant at the exit of the flow domain. For the nominally two-dimensional case using the steady RANS solver, inviscid walls were imposed for the span-wise boundaries whereas periodic boundary conditions were used for the three-dimensional unsteady simulation. The turbulent freestream intensity, T u, and the dissipation length scale, l dis , were set to 1.00% and 0.06, respectively. The DES coefficient used was that suggested by the original authors of a value of 0.6 3 .
The temporal periodicity and unsteady behavior was initially studied for the three-dimensional DES case. The information obtained was then used to determine the number of time-steps necessary to resolve a minimum of 10 periodic cycles for the time-averaged DES at the given global time-step. Figures 5 and 6(a) show the signal history for the instantaneous x-, y-, and, z-surface forces as well as the Power Spectral Density (PSD) as a function of the number of time-steps. The PSD of surface forces was plotted here as a function of the number of time-steps to more readily see periodicity in the flow. In Fig. 6(a), we see that the peak spectral signal is repeated approximately every 400 time-steps, which corresponds to a frequency of 420 Hz. The corresponding Strouhal number based on the hump height and freestream speed is approximately 0.65. Once the period was determined, this case was run for approximately 10 periodic cycles and time-averaged.   A comparison of the time-averaged skin-friction coefficient, C f , is shown in Fig. 8. Once again, there is good agreement with the steady RANS prediction upstream of the hump to x/c = 0 and over the hump to the beginning of the separation region x/c = 0.65. The DES case slightly under predicts the skin friction in the acceleration region of the flow with a max value of approximately 0.005. Similar results were shown by Morgan et. al. 23 using an implicit large-eddy simulation. Both simulations accurately predicts the onset of the separation region at approximately x/c = 0.65. In the separation region, we see the predicted reattachment point for the RANS case is at x/c = 1.2 and the time-averaged DES at x/c = 1.105. The DES matches the experimental data in the separation region much better then the RANS procedure. Table 2 shows an overview of the separation and reattachment locations for the RANS and DES procedures compared to the experimental data. Here, we can see that both procedures accurately predict the onset of separation with the RANS procedure over-predicting the size of the separation bubble. The DES, however, properly predicts the reattachment point and the separation bubble size.    Figure 9: Comparison of U-velocity contours Figure 9 shows a stream trace velocity comparison. Here we can more easily see the flow reattachment locations and the effect the DES has on the flow. In Fig. 9(b) we see that the streamline plot clearly shows that the RANS solution has a longer separation bubble than that observed experimentally in Fig. 9(a). The time-averaged DES streamlines, Fig. 9(c), show significant improvement for the predicted separation bubble length, with a 13.65% shorter mean separation region compared to RANS and a mean reattachment length that is in good agreement with experimental measurements.
Time-averaged velocity profiles at x/c = 1.0, 1.1, 1.2, and 1.3, corresponding to locations within the separation region and slightly downstream of it, were also compared with experimental data. The peak reverse flow velocity predicted at x/c = 1.0 in Fig. 10 Figure 12 shows contour plots comparing the span-wise spatially averaged vorticity and for the RANS and instantaneous DES. The instantaneous vorticity contours in Fig. 12(b) show a large range of resolved eddies consistent with large-eddy simulation treatment. Instantaneous span-wise vorticity contour slices at y = 5.2913, 5.6693, and 7.5590 cm. normal to the wall are shown in Fig. 13. Here, we can more easily see the three-dimensionality that has formed in the wake of the wall mounted hump. Figure 14 presents instantaneous vorticity isosurfaces for the DES prediction. Here we can clearly see the separated shear layer downstream of the hump.   The detached-eddy simulation procedure, described by Bush and Mani 3 , has been shown to be consistently more accurate than standard RANS. It predicts well the experimentally measured flow quantities such as pressure coefficient, surface skin friction, reattachment length, and mean velocity profiles. The RANS procedure predicted a delayed reattachment point which indicates reduced turbulent mixing inside the separation region. Attempts at using higher-order numerical techniques, applied to RANS procedures 6,8 , have shown similar results. It should be noted that the DES procedure used a single constant model coefficient of 0.6; although additional computations using various coefficients would be essential to further examine the DES procedure, it was beyond the scope of the current investigation.

V. Conclusion
A general Reynolds-averaged/detached-eddy simulation procedure was applied to the prediction of the flow over a wall-mounted hump. The initial results using the RANS and DES procedures compared well with experiment as well as other participants of NASA's 2004 Computational Fluid Dynamics Validation Workshop. Like other participants using RANS models, the onset of separation was accurately predicted the reattachment point was over-predicted. The RANS procedure also over-predicted the mean pressure, skin friction, and velocity profiles in the separation zone. The DES procedure using the Bush and Mani model showed much better results. The three-dimensional structures resolved in the wake of the DES improved the local flow physics in the separation region and the predictions of the mean pressure distribution, skin friction, and streamwise velocity.