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TECHNICAL PAPERS

On the Use of High-Order Accurate Solutions of PNS Schemes as Basic Flows for Stability Analysis of Hypersonic Axisymmetric Flows

[+] Author and Article Information
Kazem Hejranfar

Aerospace Engineering Department, Sharif University of Technology, 11365-8639, Tehran, Irankhejran@sharif.edu

Vahid Esfahanian

Mechanical Engineering Department, University of Tehran, 11365-4563, Tehran, Iranevahid@ut.ac.ir

Hossein Mahmoodi Darian

Mechanical Engineering Department, University of Tehran, 11365-4563, Tehran, Iranhmahmoodi@ut.ac.ir

J. Fluids Eng 129(10), 1328-1338 (May 15, 2007) (11 pages) doi:10.1115/1.2776962 History: Received February 20, 2006; Revised May 15, 2007

High-order accurate solutions of parabolized Navier–Stokes (PNS) schemes are used as basic flow models for stability analysis of hypersonic axisymmetric flows over blunt and sharp cones at Mach 8. Both the PNS and the globally iterated PNS (IPNS) schemes are utilized. The IPNS scheme can provide the basic flow field and stability results comparable with those of the thin-layer Navier–Stokes (TLNS) scheme. As a result, using the fourth-order compact IPNS scheme, a high-order accurate basic flow model suitable for stability analysis and transition prediction can be efficiently provided. The numerical solution of the PNS equations is based on an implicit algorithm with a shock fitting procedure in which the basic flow variables and their first and second derivatives required for the stability calculations are automatically obtained with the fourth-order accuracy. In addition, consistent with the solution of the basic flow, a fourth-order compact finite-difference scheme, which does not need higher derivatives of the basic flow, is efficiently implemented to solve the parallel-flow linear stability equations in intrinsic orthogonal coordinates. A sensitivity analysis is also conducted to evaluate the effects of numerical dissipation and grid size and also accuracy of computing the basic flow derivatives on the stability results. The present results demonstrate the efficiency and accuracy of using high-order compact solutions of the PNS schemes as basic flow models for stability and transition prediction in hypersonic flows. Moreover, indications are that high-order compact methods used for basic-flow computations are sensitive to the grid size and especially the numerical dissipation terms, and therefore, more careful attention must be kept to obtain an accurate solution of the stability and transition results.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Marching procedure and initial condition for starting the PNS solution over a blunt cone

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Figure 2

Comparison of spatial growth rates by solution of stability equations using Hermitian and Euler–Maclaurin methods for the STDS blunt-cone case, M∞=8 and Re∞=31,250 at S=175

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Figure 3

Comparison of spatial growth rates for second-order central and fourth-order compact PNS solutions for the STDS blunt-cone case, M∞=8 and Re∞=31,250 at S=175

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Figure 4

Comparison of the generalized inflection profiles G for second-order central and fourth-order compact IPNS solutions for the STDS blunt-cone case, M∞=8 and Re∞=31250 at S=175

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Figure 5

Effect of grid size and numerical dissipation value on spatial growth rates in the fourth-order compact PNS solution for the STDS blunt-cone case, M∞=8 and Re∞=31,250 at S=175

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Figure 6

Effect of accuracy of computing basic-flow derivatives on spatial growth rates in the fourth-order compact PNS solution for the STDS blunt-cone case, M∞=8 and Re∞=31,250 at S=175

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Figure 7

Comparison of spatial growth rates for fourth-order compact PNS and IPNS solutions with second-order TLNS solution for the STDS blunt-cone case, M∞=8 and Re∞=31,250 at S=175

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Figure 9

Comparison of spatial growth rates for second-order central and fourth-order compact PNS solutions for the STDS sharp-cone case, M∞=8 and Re∞=6.525×106 at Ree,l=1.73×103

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Figure 10

Comparison of spatial growth rates for the STDS sharp-cone case, M∞=8 and Re∞=6.525×106 at Ree,l=1.73×103

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Figure 11

Comparison of f=const curves of spatial growth rates for second-order central and fourth-order compact IPNS models for the STDS blunt-cone case, M∞=8 and Re∞=31,250

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Figure 12

Comparison of computed second-mode N-factor for second-order central and fourth-order compact IPNS models for the STDS blunt-cone case, M∞=8 and Re∞=31,250

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Figure 13

f=const curves of spatial growth rates for fourth-order compact PNS and IPNS models for the STDS blunt-cone case, M∞=8 and Re∞=31,250(Δf*=5kHz)

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Figure 14

Second-mode N-factor computation for fourth-order compact PNS and IPNS models for the STDS blunt-cone case, M∞=8 and Re∞=31,250(Δf*=5kHz)

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Figure 8

Comparison of spatial growth rates by solution of stability equations using Hermitian and Euler–Maclaurin methods for the STDS sharp-cone case, M∞=8 and Re∞=6.525×106 at Ree,l=1.73×103

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