^{1,a)}, D. Barkley

^{1,b)}and H. M. Blackburn

^{2,c)}

### Abstract

Results are presented from a numerical study of transient growth experienced by infinitesimal perturbations to flow in an axisymmetric pipe with a sudden 1–2 diametral expansion. First, the downstream reattachment point of the steady laminar flow is accurately determined as a function of Reynolds number and it is established that the flow is linearly stable at least up to . A direct method is used to calculate the optimal transient energy growth for specified time horizon , Re up to 1200, and low-order azimuthal wavenumber . The critical Re for the onset of growth with different is determined. At each Re the maximum growth is found in azimuthal mode and this maximum is found to increase exponentially with Re. The time evolution of optimal perturbations is presented and shown to correspond to sinuous oscillations of the shear layer. Suboptimal perturbations are presented and discussed. Finally, direct numerical simulation in which the inflow is perturbed by Gaussian white noise confirms the presence of the structures determined by the transient growth analysis.

We are grateful to Edward Hall for sharing results from eigenvalue calculations on the expanding-pipe flow and to Andrew Cliffe for discussions about the implications of the eigenvalue calculation for experiments. Computing facilities were provided by the UK Centre for Scientific Computing of the University of Warwick with support from the Science Research Investment Fund, and by the Australian National Computational Infrastructure National Facility, through MAS Grant No. D77.

I. INTRODUCTION

II. METHODOLOGY

A. Governing equations and flow geometry

B. Linear stability and transient growth problems

C. Further details

III. RESULTS

A. Base flows

B. Linear stability

C. Transient energy growth

1. Dependence on azimuthal mode number

2. Reynolds number and time horizon

3. Time evolution of optimal perturbations

4. Growth maxima

5. Suboptimal growth

D. Response to noise

IV. SUMMARY AND DISCUSSION

### Key Topics

- Reynolds stress modeling
- 38.0
- Eigenvalues
- 29.0
- Separated flows
- 20.0
- Flow instabilities
- 17.0
- Normal modes
- 13.0

## Figures

Sketch illustrating the evolution of a perturbation through an expanding pipe. Small inlet perturbations are amplified in the region of the separated axisymmetric shear layer, but eventually decay downstream. Hence, even though the flow is linearly stable, it supports very strong transient growth of perturbations.

Sketch illustrating the evolution of a perturbation through an expanding pipe. Small inlet perturbations are amplified in the region of the separated axisymmetric shear layer, but eventually decay downstream. Hence, even though the flow is linearly stable, it supports very strong transient growth of perturbations.

Geometry of the expanding pipe. The computational flow domain is illustrated with the cylindrical coordinate system and the inlet and outlet lengths indicated (not to scale).

Geometry of the expanding pipe. The computational flow domain is illustrated with the cylindrical coordinate system and the inlet and outlet lengths indicated (not to scale).

A spectral-element mesh used in this study. For the case illustrated there are 563 elements, an inflow length of , and an outflow length of . When required, meshes with more elements and outflow lengths up to have been used. The two-dimensional mesh (a) used for linear analysis is extended to three dimensions (b) for nonlinear analysis (DNS) where Fourier expansions are used in azimuth.

A spectral-element mesh used in this study. For the case illustrated there are 563 elements, an inflow length of , and an outflow length of . When required, meshes with more elements and outflow lengths up to have been used. The two-dimensional mesh (a) used for linear analysis is extended to three dimensions (b) for nonlinear analysis (DNS) where Fourier expansions are used in azimuth.

Contour plot of the base-flow streamfunction at , showing the separation and reattachment of the flow, and the recirculation region behind the expansion. Contours are drawn at intervals of 0.125 in the core of the flow and at intervals of 0.02 in the recirculation region.

Contour plot of the base-flow streamfunction at , showing the separation and reattachment of the flow, and the recirculation region behind the expansion. Contours are drawn at intervals of 0.125 in the core of the flow and at intervals of 0.02 in the recirculation region.

Relationship between downstream reattachment point and Reynolds number for base flows with a fully developed inlet profile up to . Points are the computed value of and the solid line shows the best-fit proportionality given by . The dotted lines indicate reattachment lengths for the base flows at (corresponding to Fig. 4) and .

Relationship between downstream reattachment point and Reynolds number for base flows with a fully developed inlet profile up to . Points are the computed value of and the solid line shows the best-fit proportionality given by . The dotted lines indicate reattachment lengths for the base flows at (corresponding to Fig. 4) and .

(a) Energy growth envelopes at for azimuthal mode numbers (as indicated) up to . (b) Enlargement of (a) for small . Curves for through are in decreasing monotonic order on the right-hand side.

(a) Energy growth envelopes at for azimuthal mode numbers (as indicated) up to . (b) Enlargement of (a) for small . Curves for through are in decreasing monotonic order on the right-hand side.

Contours of optimal transient energy growth as a function of time horizon and Reynolds number for (a) azimuthal mode number and (b) azimuthal mode number .

Contours of optimal transient energy growth as a function of time horizon and Reynolds number for (a) azimuthal mode number and (b) azimuthal mode number .

Energy growth under linear evolution at , for three different initial conditions corresponding to optimal perturbations at , 70, and 110. Circles denote the optimal growth envelope. Each linear evolution curve touches the envelope at its respective value, as indicated by a filled circle.

Energy growth under linear evolution at , for three different initial conditions corresponding to optimal perturbations at , 70, and 110. Circles denote the optimal growth envelope. Each linear evolution curve touches the envelope at its respective value, as indicated by a filled circle.

Evolution of optimal initial disturbance in the mode visualized through contours/isosurfaces of azimuthal velocity at from (bottom) in time intervals of four units in the spatial range of . The panel labeled shows the evolved disturbance at its maximum growth: here the spatial range is and the isosurface levels are two orders of magnitude larger than in the other panels.

Evolution of optimal initial disturbance in the mode visualized through contours/isosurfaces of azimuthal velocity at from (bottom) in time intervals of four units in the spatial range of . The panel labeled shows the evolved disturbance at its maximum growth: here the spatial range is and the isosurface levels are two orders of magnitude larger than in the other panels.

Evolution of optimal initial disturbance in the mode visualized through isosurfaces of radial velocity at from (bottom) in time intervals of two units in the spatial range of .

Evolution of optimal initial disturbance in the mode visualized through isosurfaces of radial velocity at from (bottom) in time intervals of two units in the spatial range of .

Physical interpretation of the maximal disturbance at . Shown is a linear superposition of the base flow with the optimal disturbance at the time of maximum growth, . The disturbance has a relative energy magnitude of 5% compared with the base flow. The visualization is a semitransparent isosurface of azimuthal vorticity at a level highlighting the separated shear layer.

Physical interpretation of the maximal disturbance at . Shown is a linear superposition of the base flow with the optimal disturbance at the time of maximum growth, . The disturbance has a relative energy magnitude of 5% compared with the base flow. The visualization is a semitransparent isosurface of azimuthal vorticity at a level highlighting the separated shear layer.

Optimal energy growth , as a function of Re for each of the first four azimuthal modes. Above , increases exponentially with Reynolds number at a rate of 0.45 orders of magnitude for each increase in 100 in Re. Inset shows detail at small Re. Curves intersect at the critical Reynolds number of each mode .

Optimal energy growth , as a function of Re for each of the first four azimuthal modes. Above , increases exponentially with Reynolds number at a rate of 0.45 orders of magnitude for each increase in 100 in Re. Inset shows detail at small Re. Curves intersect at the critical Reynolds number of each mode .

Location of the centroid of the disturbance at its maximum growth (points connect by solid lines), compared to the location of the reattachment point (dashed line). The disturbance reaches its maximum approximately five diameters upstream of reattachment point.

Location of the centroid of the disturbance at its maximum growth (points connect by solid lines), compared to the location of the reattachment point (dashed line). The disturbance reaches its maximum approximately five diameters upstream of reattachment point.

Leading four growth envelopes for the optimal and suboptimal perturbations for the mode at . The circles on the dotted line at show the first eight leading eigenvalues at the point of the peak optimal growth.

Leading four growth envelopes for the optimal and suboptimal perturbations for the mode at . The circles on the dotted line at show the first eight leading eigenvalues at the point of the peak optimal growth.

Isosurfaces of azimuthal velocity for the four leading perturbations at , evolved to time . (a) and (b) are the optimal and first suboptimal modes. (c) and (d) are the next pair of suboptimal modes (see Fig. 14).

Isosurfaces of azimuthal velocity for the four leading perturbations at , evolved to time . (a) and (b) are the optimal and first suboptimal modes. (c) and (d) are the next pair of suboptimal modes (see Fig. 14).

Modal energy in a noisy inflow DNS of the expanding-pipe flow at (a) , (b) , and (c) . Modal energies visible above the noise floor of are labeled. Initially the flow is the steady axisymmetric base flow. The axisymmetric energy is off the scale of the figure.

Modal energy in a noisy inflow DNS of the expanding-pipe flow at (a) , (b) , and (c) . Modal energies visible above the noise floor of are labeled. Initially the flow is the steady axisymmetric base flow. The axisymmetric energy is off the scale of the figure.

Visualization of noise-driven flow and optimal perturbation at (a) and (b) in the range of . The upper half of each pipe shows the optimal perturbation at the respective Reynolds number, evolved to the point of maximum growth. The lower half shows a snapshot of the noise-driven flow for comparison.

Visualization of noise-driven flow and optimal perturbation at (a) and (b) in the range of . The upper half of each pipe shows the optimal perturbation at the respective Reynolds number, evolved to the point of maximum growth. The lower half shows a snapshot of the noise-driven flow for comparison.

Isosurfaces of streamwise (top), radial (middle) and azimuthal (bottom) velocity components for the noise-driven flow showing the downstream disturbance induced by the stochastic forcing at the same time instant as Fig. 17(b).

Isosurfaces of streamwise (top), radial (middle) and azimuthal (bottom) velocity components for the noise-driven flow showing the downstream disturbance induced by the stochastic forcing at the same time instant as Fig. 17(b).

Energy of noise-driven flow through the line for the (solid lines) and (dotted lines). [(a)–(c)] , (d) , and (e) at 1/4 the noise level of (d). Vertical lines indicate , the centroid of the optimal linear perturbation at the corresponding value of Re.

Energy of noise-driven flow through the line for the (solid lines) and (dotted lines). [(a)–(c)] , (d) , and (e) at 1/4 the noise level of (d). Vertical lines indicate , the centroid of the optimal linear perturbation at the corresponding value of Re.

Radial profiles of velocity component standard deviations comparing the noise-driven simulation (left) with the linear analysis (right). The three velocity components each normalized to their peak value are shown: streamwise (solid line), radial velocity (dashed line), and azimuthal velocity (dotted line).

Radial profiles of velocity component standard deviations comparing the noise-driven simulation (left) with the linear analysis (right). The three velocity components each normalized to their peak value are shown: streamwise (solid line), radial velocity (dashed line), and azimuthal velocity (dotted line).

Dependence of maximum transient energy growth on bulk Reynolds numbers for three separated flows: two-dimensional disturbances in a backward-facing step flow (Ref. 23); steady stenotic flow (Ref. 9); and the present 1–2 axisymmetric expansion, compared with Hagen–Poiseuille flow (Refs. 44 and 47).

Dependence of maximum transient energy growth on bulk Reynolds numbers for three separated flows: two-dimensional disturbances in a backward-facing step flow (Ref. 23); steady stenotic flow (Ref. 9); and the present 1–2 axisymmetric expansion, compared with Hagen–Poiseuille flow (Refs. 44 and 47).

## Tables

Effect of domain length on representative quantities at . The reattachment point, leading eigenvalue, and optimal growth for are given for different inflow and outflow lengths. The polynomial order is .

Effect of domain length on representative quantities at . The reattachment point, leading eigenvalue, and optimal growth for are given for different inflow and outflow lengths. The polynomial order is .

Streamwise position of base flow reattachment point (in units of step height, ) at , as a function of spectral-element polynomial order . The outflow length is .

Streamwise position of base flow reattachment point (in units of step height, ) at , as a function of spectral-element polynomial order . The outflow length is .

Leading eigenvalues from a linear stability analysis of flows in the 1–2 axisymmetric expansion for Reynolds numbers indicated. All values correspond to azimuthal mode number . All are real and negative and hence the axisymmetric base flow is linearly stable up to at least .

Leading eigenvalues from a linear stability analysis of flows in the 1–2 axisymmetric expansion for Reynolds numbers indicated. All values correspond to azimuthal mode number . All are real and negative and hence the axisymmetric base flow is linearly stable up to at least .

Table of critical Reynolds numbers , maximum growth values , and corresponding time horizons for maximum growth, (for and as indicated) for each of the first four azimuthal modes, .

Table of critical Reynolds numbers , maximum growth values , and corresponding time horizons for maximum growth, (for and as indicated) for each of the first four azimuthal modes, .

Characteristics of the optimal perturbations at the time of maximum growth for modes at Reynolds numbers indicated. Along with the centroid location , of the evolved perturbation energy, as well as the local axial wavelength and temporal frequency St of the perturbation.

Characteristics of the optimal perturbations at the time of maximum growth for modes at Reynolds numbers indicated. Along with the centroid location , of the evolved perturbation energy, as well as the local axial wavelength and temporal frequency St of the perturbation.

Statistics for DNS with stochastic inflow forcing at . Here we define the centroid based on the distribution of turbulent kinetic energy in the outflow section of the pipe.

Statistics for DNS with stochastic inflow forcing at . Here we define the centroid based on the distribution of turbulent kinetic energy in the outflow section of the pipe.

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