^{1,a)}and Sanjiva K. Lele

^{2}

### Abstract

A two-dimensional model problem of sound generation due to prescribed unsteady starting and stopping motion of a circular cylinder is studied by the numerical solution of two-dimensional compressible Navier–Stokes equations. The unsteady flow near the cylinder surface and the sound generation and propagation are analyzed with respect to three parameters: the time scale of the startup and stopping motion, the peak Mach number (based on peak velocity and ambient speed of sound), and the Reynolds number (based on peak velocity, cylinder diameter, ambient density, and dynamic viscosity). The flow behavior is studied for fast (time scale of unsteady motion similar to acoustic time scale) or very slow startup motion. The Mach number is varied between 0.1 and 0.4 and the Reynolds number between 150 and 9500. The accuracy of unsteady flow solution is demonstrated by comparison to the experimental data. For fast startup motion, a sharp peak is observed in the drag curve during the acceleration phase of motion. We find that the dominant contribution to total drag comes from the pressure drag and the peak drag force scales with peak velocity. The startup process introduces weak shock waves in the flow, which contributes to the far-field noise. For slow startup motion, the drag curve is smoother and the drag force scales with the square of the peak velocity. For fast startup motion, the acoustic energy that propagates to the far-field scales with the square of the peak velocity, whereas for slow startup, it scales with the fourth power of the peak velocity. The amount of sound energy radiated to the far-field is found to be a small percentage of the total energy input to set up the entire motion of cylinder. An interpretation for the observed scaling relations in the numerical data is presented using different asymptotically valid reduced forms of the governing equations.

This work was supported in part by a grant from the Supersonics Element of the Fundamental Aeronautics Program of NASA under NRA Grant No. NNX07AC94A. Technical help from Dr. S. Nagarajan and Dr. R. Bhaskaran is gratefully appreciated. The authors would also like to thank the anonymous referees for their helpful comments.

I. INTRODUCTION

II. NUMERICAL TECHNIQUE

III. RESULTS

A. Unsteady flow in the near-field

B. Energy budget

IV. SCALING ANALYSIS OF THE UNSTEADY RESPONSE

V. SUMMARY

### Key Topics

- Viscosity
- 26.0
- Kinematics
- 25.0
- Reynolds stress modeling
- 20.0
- Rotating flows
- 15.0
- Acoustic waves
- 14.0

## Figures

Schematic of the coordinate system used. Direction of motion of the cylinder is in the direction.

Schematic of the coordinate system used. Direction of motion of the cylinder is in the direction.

Prescribed velocity to the cylinder as a function of time. Shown for the case and .

Prescribed velocity to the cylinder as a function of time. Shown for the case and .

Contours of vorticity and dilatation shown for , , and . Both are nondimensionalized by : (a) , (b) , (c) , and (d) . Vorticity contours in color from −1 (blue) to 1 (red) in steps of 0.02. Dilatation (negative in solid line and positive in dotted line) plotted from −0.1 to 0.1 in steps of 0.025.

Contours of vorticity and dilatation shown for , , and . Both are nondimensionalized by : (a) , (b) , (c) , and (d) . Vorticity contours in color from −1 (blue) to 1 (red) in steps of 0.02. Dilatation (negative in solid line and positive in dotted line) plotted from −0.1 to 0.1 in steps of 0.025.

Stream traces behind the cylinder at various Reynolds numbers. (a) Bulging of stream traces shown for case B. , , and . (b) Formation of an isolated vortex, the same case as in (a) and . (c) Vortex structure for case C: , , and . (d) Vortex structure for case D: , , and .

Stream traces behind the cylinder at various Reynolds numbers. (a) Bulging of stream traces shown for case B. , , and . (b) Formation of an isolated vortex, the same case as in (a) and . (c) Vortex structure for case C: , , and . (d) Vortex structure for case D: , , and .

Variation of streamwise velocity along the axis of motion on the rear side at various time instants. Solid line: present calculation; symbols: experimental data of Bouard and Coutanceau (Ref. 14): (△) , (○) , and . (a) for , , and . (b) for , , and .

Variation of streamwise velocity along the axis of motion on the rear side at various time instants. Solid line: present calculation; symbols: experimental data of Bouard and Coutanceau (Ref. 14): (△) , (○) , and . (a) for , , and . (b) for , , and .

Evolution of drag coefficients with time for different values of but fixed and . Total drag: solid line; pressure drag: dashed line; viscous drag: dashed dotted line. (a) (b) .

Evolution of drag coefficients with time for different values of but fixed and . Total drag: solid line; pressure drag: dashed line; viscous drag: dashed dotted line. (a) (b) .

Variation of peak pressure drag and peak total drag . (a) Variation with for , . Total drag: -△- ; pressure drag: -○-. (b) Variation with for and . Total drag: -△-; pressure drag: -○-. (c) Variation of total drag with . , : -△-; , : -◻-; and , : . Total drag for , : . Dotted line: Reference line with slopes 1 and 2.

Variation of peak pressure drag and peak total drag . (a) Variation with for , . Total drag: -△- ; pressure drag: -○-. (b) Variation with for and . Total drag: -△-; pressure drag: -○-. (c) Variation of total drag with . , : -△-; , : -◻-; and , : . Total drag for , : . Dotted line: Reference line with slopes 1 and 2.

Wall vorticity at different time instants for , , and . : dotted line; : dashed line; : -△-; : dashed dotted line; : -◻-; and : solid line.

Wall vorticity at different time instants for , , and . : dotted line; : dashed line; : -△-; : dashed dotted line; : -◻-; and : solid line.

(a) Normal pressure gradient at the wall at different time instants; (b) Tangential pressure gradient at the wall at different time instants. Legend: the same as in Fig. 8.

(a) Normal pressure gradient at the wall at different time instants; (b) Tangential pressure gradient at the wall at different time instants. Legend: the same as in Fig. 8.

Comparison of wall vorticity for different at different time instants. , , and : dotted line; , , and : dashed line; , , and : solid line; , , and : dashed dotted line. (a) at and (b) at .

Comparison of wall vorticity for different at different time instants. , , and : dotted line; , , and : dashed line; , , and : solid line; , , and : dashed dotted line. (a) at and (b) at .

Contours of density, , plotted between −0.1 and 0.1 are shown in column (i). Contours of energy components, , , and plotted between and 0.03 are shown in columns (ii)–(iv), respectively. Row (a) corresponds to , (b) to , and (c) to . Shown for case , , and .

Contours of density, , plotted between −0.1 and 0.1 are shown in column (i). Contours of energy components, , , and plotted between and 0.03 are shown in columns (ii)–(iv), respectively. Row (a) corresponds to , (b) to , and (c) to . Shown for case , , and .

(a) Energy budget implied by time integrated form of Eq. (9) is shown for cases with , , (upper curve) and 3000 (lower curve). : solid line; other terms in Eq. (9): dashed dotted line. (b) Evolution of acoustic and entropy energy fluxes with time is shown for , , and two values of . for : solid line; for : dashed line; for : -△-; for : -◻-. (c) Evolution of kinetic , potential , and entropy energies with time is shown for , , and two values of . for : solid line; for : dashed line; for : -△-; for : -◻-; for : dashed dotted line; for : dotted line. All calculations are shown for control surface location .

(a) Energy budget implied by time integrated form of Eq. (9) is shown for cases with , , (upper curve) and 3000 (lower curve). : solid line; other terms in Eq. (9): dashed dotted line. (b) Evolution of acoustic and entropy energy fluxes with time is shown for , , and two values of . for : solid line; for : dashed line; for : -△-; for : -◻-. (c) Evolution of kinetic , potential , and entropy energies with time is shown for , , and two values of . for : solid line; for : dashed line; for : -△-; for : -◻-; for : dashed dotted line; for : dotted line. All calculations are shown for control surface location .

Variation of total acoustic energy, , in the far-field. (a) Variation with for and . (b) Variation with for , (lower curve) and 0.3 (upper curve). (c) Scaling with . , : -○-; , : -◻-; , : -△-; , : -◁-; , : dashed dotted line with circles. Dotted line: Reference lines of slope 2 and 4.

Variation of total acoustic energy, , in the far-field. (a) Variation with for and . (b) Variation with for , (lower curve) and 0.3 (upper curve). (c) Scaling with . , : -○-; , : -◻-; , : -△-; , : -◁-; , : dashed dotted line with circles. Dotted line: Reference lines of slope 2 and 4.

Energy balance shown for control surface location . Legend: the same as in Figs. 12(b) and 12(c). (a) Surface integral terms. (b) Volume integral terms.

Energy balance shown for control surface location . Legend: the same as in Figs. 12(b) and 12(c). (a) Surface integral terms. (b) Volume integral terms.

## Tables

Mesh parameters for conditions of the simulations presented.

Mesh parameters for conditions of the simulations presented.

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