The objective of the present study was to establish grid and time step independence and to determine the computational efficiency of EXN/Aero in transient simulations. In the following the term solution convergence will be reserved for the progress towards grid and time step independence. For this purpose nine simulations were run with different time step sizes and mesh sizes. These nine cases were spanned by, , 11 M and 22 M node meshes.
The subject of the present study is the accelerating cylinder problem which has been studied by analytical methods  as well as experimentally and computationally. The cylinder considered here is modelled on a companion experiment conducted by Envenio. It has a diameter, and length, , and travels a maximum distance of during the transient motion.
The direction of the lift and drag forces are shown in Figure 1.The experiment and simulations were performed for water with and . The maximum Reynolds number obtained by the cylinder is, .
An important point of comparison for the force on a cylinder accelerating from rest in an infinite medium is the potential flow result
and due to symmetry of the flow. This result will be used in the following to evaluate the CFD simulation results during the early phase of motion.
The nominal acceleration of the cylinder was, with measured velocity and displacement of the transient cylinder motion shown in Figure 2. The corresponding acceleration is shown in Figures 3 and 4. These kinematics represent the ensemble average of 20 repetitions of the experiment.
The convection time defined as the time required for the cylinder to move one diameter at its maximum velocity is
The ratio of the 3 integration time steps to the convective time step are .
Figure 1: Schematic of accelerating cylinder showing lift and drag directions.
Figure 2: Ensemble average kinematics for cylinder motion based on 20 repetitions of the experiment. The velocity data (indicated by the symbols) forms a table of prescribed motion input to EXN/Aero. The sold line represents the interpolated velocity used at each time step. Interpolation was done using cubic splines.
Figure 3: Acceleration curve for the entire transient event derived from the velocity curve shown in Figure 2. The acceleration is used to calculate the body forces in the non-inertial frame of reference.
Figure 4: Acceleration during the early phase of the transient event derived from the velocity data of Figure 2.
2 Computational Domain and Boundary Conditions
The computational domain used in the present study is shown in Figure 5. Its geometry is similar to that of the tow tank in which the experiments were performed. The computations are be performed in the accelerating frame of the cylinder.
The boundary conditions on the top and sides of the domain were taken to be symmetry planes without shear stresses. The bottom surface of the domain had the velocity set to the negative of the cylinder velocity, to correctly model its relative motion. The exit plane was specified as a constant pressure surface. The boundary values: and giving mm.
Figure 5: Geometry of the computational domain located in the cylinder frame of reference. Inlet and side views shown.
A structured mesh having 5.5 million nodes is shown in Figure 6. The first node near the cylinder surface had and double precision was used in the 4 blocks near the cylinder surface. Two finer meshes were obtained by reducing the mesh spacing by in along each coordinate successively. Time stepping used backward second order differencing and differencing of the advection terms was done using a second order upwind biased TVD scheme. Simulations were run using SST-DES turbulence model option of EXN/Aero. A summary of the wall clock time per time step and total simulation time for the 9 cases is provided in Table 1.
Figure 6: Top and side views of the computational mesh used for the present simulations. Total node count 5.5 million. Finer meshes of 11 M and 22 M nodes were obtained by reducing the mesh spacing uniformly in all directions by successively.
3. Computational Results
The simulation results for the lift and drag force over the early phase of the cylinder motion where are shown in Figure 7. The plots are organized to show the effect of time step on the solution. The potential flow result described by equation (1) (based on the same acceleration history using in the EXN/Aero solution) is shown for comparison. In the early phase of motion the EXN/Aero solution converges to the potential flow result with mesh and time step refinement.
Figure 7: The effect of the time step on the drag force during the early phase of the motion. The dashed line is the force according to a potential flow solution corresponding to the acceleration curve shown in Figure 3.
We now consider the motion of the forces acting on the cylinder in the time interval, , during which the cylinder has a substantial forward velocity and flow separation is well developed. Figure 8 shows the development of the vorticity on the mid-plane of the cylinder.
Figure 9 shows the effect of time step on the lift and drag force. The solution converges everywhere with increased temporal resolution uniformly. Figure 10 compares the drag forees for the combinations of mesh size and time step along the diagonal of Table 1. These simulations have a resolution which has a fixed product, , where is the mesh node count. The force on the accelerating cylinder for a potential flow is shown for comparison.
Figure 8: The development of the vorticity field in increments starting at . The cylinder begins decelerating at .
Figure 9: The effect of time step on the lift and drag force during the viscous phase of the transient.
Figure 10: Comparison of the solutions with time steps and mesh sizes on the diagonal of Table 1.
We can define the viscous contribution of the drag force, , on the cylinder as that over and above the drag force resulting from the unsteady potential flow force, , calculated from equation (1)
A plot of the viscous drag force derived from the simulations is shown in Figure 11. The drag coefficient for , is shown on the secondary ordinate. The viscous contribution has a period of adjustment spanning a time interval, , after which it remains almost constant until the cylinder decelerates. At which point reduces sharply. Once the cylinder has stopped the viscous force changes sign before gradually diminishing to zero. The drag coefficient is lower than the accepted value for cylinder in steady cross flow Ref  at the same Reynolds number, . A result consistent with previous studies.
Figure 11: The viscous contribution to the drag force, , as defined by equation (3).
Time histories of the surface shear stress at specific angular positions on the mid plane normal to the cylinder axis are shown in Figures 12 and Figure 13. The angular position is measured from the leeward side as illustrated in Figure 1 and the shear stress is considered negative in the direction of cylinder motion. Figure 12 shows that the separation begins in the vicinity of and then spreads as far forward as before the cylinder begins decelerating. Once the deceleration begins the shear stress is reversed as far forward .
Figure 13 shows the surface shear stress in the wake region. Even in the interval where the cylinder is moving at near constant velocity the surface shear stress in this region is complex and includes the development of flow re-attachment and secondary separation. Also shown are the shear stress time histories for symmetrical positions on the surface. The symmetry is maintained up to after which asymmetry and the magnitude of the shear stress become very substantial during the interval of deceleration. These results can be compared to the vorticity plots in Figure 8.
Figure 12: Surface shear stress time histories at angular positions around the surface of the cylinder forward of the wake region.
Figure 13: Surface shear stress time histories at angular positions around the cylinder in the wake region. Dashed lines represent symmetrical positions on the cylinder surface to solid lines.
- Karamcheti, K, Principles of Ideal-Fluid Aerodynamics, Krieger, 1980.