Three-Dimensional Finite Difference Method for Rotordynamic Fluid Forces on Seals

Abstract

does not predict a streamwise stress level that is signi® cantly higher than the simulation of Beaudan and Moin. 5 However, this can be explainedby noting that, for this¯ow, most of the contributionto the Reynolds stress comes from¯uctuationsin a narrow frequencyband extending from about 0.5x s t to 3.0x st , and in this frequency band, the energy in both simulations is comparable. Thus, even though the simulations of Beaudan and Moin 5 exhibit signi® cant damping of the higher frequencies,this does not have a signi® cant impact on the low-order turbulence statistics.By comparing the vertical stress pro® les at these locations, we observe that prediction from the two simulations at x/ D = 7.0 is quite similar. At x/ D = 10.0, the two simulations predict roughly the same peak stress level; however, the shape of the experimental pro® le matches the pro® le of Beaudan and Moin 5 better than it does for the current simulation. Furthermore, we have found that vertical velocity and shear stress pro® les (not shown here) from both the simulations are also in reasonable agreement with experiments. 11 ConclusionsIt is found that in the downstream portion of the wake, where the grid is relatively coarse, the numerical dissipation inherent in the higher-order upwind-biased schemes removes substantial energy from roughly three-quarters of the resolved wave number range. In the central difference simulation, because there is no numerical dissipation, the smaller scales are more energetic. Because of this reduction in the damping of smaller scales, we ® nd that the computed power spectra agree well with the experiment up to about half of the resolved wave number range. However, the enhanced energy in the small scales has no signi® cant effect on the low-order statistics, and the mean velocity and Reynolds stress pro® les in this region obtained from the two simulations are comparable. This is because most of the contribution to the normal stress comes from uctuations whose frequency is centered in a narrow band around the shedding frequency and change in the energy of the small scales does not have a signi® cant effect on the magnitudes of the Reynolds stresses. However, in applications such as¯ow generated noise and reactive¯ows, small-scale¯uctuations play a crucial role, and it is, therefore, critical to retain the energy in the small scales. In such applications,energy conservativeschemes would be preferableover upwind schemes. We also ® nd that with about 20±30% smaller grid spacing, the second-order central difference scheme gives results that are comparable to those obtained by the high-order upwind biased schemes. The higher-order upwind based solver is more expensive on a per-pointbasis than the second-ordercentral difference solver, and this partially offsets the additional cost of the increased resolution required by the second-order method. A drawback of the second-order central scheme is that the simulations are sensitive to numerical factors such as grid discontinuities and out¯ow boundary conditions and, thus, g...

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