TRANSIENT GROWTH: EXPERIMENTS, DNS AND THEORY

F. Gökhan Ergin
Mechanical and Aerospace Engineering Department,
Case Western Reserve University
Cleveland, Ohio 44106, USA
gokhan@case.edu

Meelan ChoudhariComputational Modelling and Simulation Branch,
NASA Langley Research Center
Hampton, Virginia 23681, USA
m.m.choudhari@larc.nasa.gov

Paul Fischer
Mathematics and Computer Science Division,
Argonne National Laboratory
Argonne, Illinois 60439, USA
fischer@mcs.anl.gov

Anatoli Tumin
Department of Aerospace and Mechanical Engineering,
The University of Arizona
Tucson, Arizona 85721, USA
tumin@eng.arizona.edu

ABSTRACTTransient growth of linearly stable disturbances is believedto play an important role in the subcritical transition of laminar boundary layers and the self-sustained nature of boundary
layer fluctuations in a fully turbulent flow. Prior work ontransient growth has focused on identifying the optimum initial disturbances that result in maximum transient growth.
This paper addresses the companion issue of receptivity of
those disturbances, the mechanism that determines the actual
magnitudes of transient growth that are realized in a given
physical situation. A synergistic combination of experimental,
computational, and theoretical approaches is used to quantify
the flow receptivity to surface roughness in a Blasius boundary layer. Results reveal the non-optimality of the transient
growth factors involved as well as the sensitive dependenceof flow perturbations to the geometric characteristics of theroughness distribution. Direct numerical simulations (DNS)
are compared in detail with experimental results, results obtained from linear receptivity theory and optimal disturbancecalculations. DNS shows good agreement with the experimental results. Differences between the linear theory and DNS are
attributed to nonlinear receptivity mechanisms. Results also
support the proposal by Fransson et al. (2004) that disagreement between optimal disturbances and experiments/DNS
may be attributed to differences involving the wall normal location of the streamwise vortex initiating the transient growth.

INTRODUCTION
The transient growth phenomenon refers to an algebraic
amplification of small-amplitude disturbances prior to an exponential decay farther downstream. Transient growth has
been proposed as a likely mechanism behind laminar-turbulent
transition scenarios that cannot be explained by the classicalparadigm of hydrodynamic instabilities. Transient growth isalso believed to play an important role in the self-generation of
turbulence in fully turbulent wall shear flows (Butler and Farrell 1993; Chernyshenko and Baig 2005). Physically, the occurrence of transient growth can be explained by Landahl’s (1980)“lift-up” mechanism, where a pair of stable, counter-rotating,streamwise-oriented vortices transfers momentum across theboundary layer, creating a significantly stronger streamwise
velocity perturbation.
The mathematical foundation of transient growth has been
described by Schmid and Henningson (2001) and Reshotko(2001). Transient growth arises because the linearized disturbance equations are not self-adjoint and, therefore, havenonorthogonal eigenmodes. In boundary layers, the correct
representation of disturbances is the superposition of the discrete and continuous eigenmodes of the Orr–Sommerfeld (OS)equation. In classical linear stability analysis, the most amplified discrete eigenmode represents the dominant disturbanceobserved in an experiment. However, even in the subcriticalregion where the TS waves are damped, a suitable mixtureof continuous spectrum modes (each of which has a differentdecay rate) can exhibit temporary algebraic growth before theexponential decay manifests itself.
A particular combination of the modes will form an initial
disturbance that experiences the maximum amount of growth.
Such a disturbance is called the optimal disturbance (Farrell1988) and many investigators have contributed to the modelling of spatially growing optimal disturbances (Andersson
et al. 1999; Luchini 2000; Tumin and Reshotko 2001). Fora laminar Blasius boundary layer, the optimal initial disturbance has been shown to consist of an array of stationary
streamwise vortices with a dimensionless spanwise wavenumber of 0.45. Andersson et al. (1999) also found that themeasured evolution of low-frequency boundary-layer disturbances excited via freestream turbulence is in agreement withthe optimal growth theory. On the other hand, there is asignificant mismatch between the optimal growth theory andmeasured (i.e. realizable) disturbances due to controlled surface roughness. White (2002) finds that roughness-induced
disturbances show suboptimal behavior. Fransson et al. (2004)
confirm this observation and attribute the mismatch to differences between the initial disturbance profiles used in optimal
disturbance studies and those induced by surface roughness.
The former corresponds to a streamwise vortex that is notconfined to the boundary layer. Unlike the case of freestream
turbulence, however, disturbances generated by low amplitude
surface roughness are mostly confined to the boundary layer
region.
Transient disturbances are extremely sensitive to initial
disturbance conditions as these determine the spectrum ofmodes that make up the disturbance and, hence, its algebraic
growth rate. Since the maximum disturbance energy attained
downstream of the source is mostly specified by the algebraic
growth rate, understanding the receptivity mechanism that
determines the initial disturbance condition is crucial. The
objective of this work is to summarize the recent progress inexperimental, computational and theoretical investigations ofroughness-induced transient growth, with an emphasis on detailed comparisons between the respective findings for specificroughness configurations. Remaining challenges to assessing
the relevance of the optimal growth theory to roughness effects
on boundary-layer transition are also outlined.
EXPERIMENTS
White and coworkers (White and Ergin 2003; White et al.
2005; Ergin and White 2005) at Case Western Reserve University (Case) have conducted a number of experiments on the
behavior of roughness-induced transient disturbances. Present
efforts are focused on generating disturbance data that is suitable for comparison with DNS and theoretical models. A brief
description of the experimental setup and data analysis procedure is given below.
Measurements are obtained in the boundary layer of a flat
plate with an elliptical leading edge and disturbances are generated by a spanwise array of cylindrical roughness elements.
Roughness arrays provide considerable flexibility in varying
the controlled disturbance inputs, via variations in the height,
spacing and diameter of the roughness elements. Various averaging techniques, such as spatial phase-locked averaging,can be utilized to improve the signal-to-noise characteristics during data analysis. Throughout this paper, k denotes
the roughness height, λk is the spanwise roughness spacing(19 mm), xk is the array’s streamwise location relative to thephysical leading edge (300 mm), and Rek is the roughness
Reynolds number, U (k)k/ν. Hotwire velocity measurements
are performed in planes perpendicular to the streamwise flow.
An illustration of the experimental setup is given in Figure 1.
The main data analysis technique is to decompose the kinetic energy associated with the steady disturbance into spanwise wavelength components by performing a spanwise Fouriertransform, computing the energy of each and examining thedownstream evolution of these energies. The energy is definedbyZ∞

Eλ (x) =PSD(x, η, λ)dη(1)0where PSD is the one-sided power spectral density.
For Rek = 202, the phase-lock averaged steady streamwise
velocity contours obtained 30 mm downstream of the roughness array are shown in Figure 2. In this figure, the flow is into
the page; the abscissa is the spanwise coordinate normalizedby λk , and the ordinate is the dimensionless wall-normal coordinate, η. The contour lines indicate 10% increments of thefreestream velocity and the box in the middle approximates
the planform shape of the cylindrical roughness element thatgenerated this disturbance. The decelerated regions along theroughness centerline and accelerated regions along the edges
agree well with previous experimental results. The measured
streamwise velocity contours can be compared to DNS results
which are shown in Fig. 3.
White and Ergin (2003) investigated how the steady disturbance energy in spatial wavelengths scales with roughness
amplitude. For Rek = 119 and a diameter of 6.35 mm (whichcorresponds to λk /3), Fig. 4 shows the downstream evolutionof the steady disturbance energy contained in λk , λk /2, λk /3,and λk /4 components. It was found that the disturbance energy approximately scales with Re2k (i.e., k 4 ), and this scaling
is further verified by White et al. (2005) for Rek ranging from
16 to 195. Rice (2004) extends the range to 254. Fig. 5 shows
the transient growth of Eλk /3 for selected values of Rek . In
this figure, the solid lines denote the least squares fits to thedata using the model function a(x − xk )exp[−(x − xk )/b].
White et al. (2005) also considered the effect of roughness diameter on disturbance energy and found that even asmall change in roughness diameter had a dramatic effect on
the disturbance spectrum, both quantitatively and qualitatively. The effect of roughness diameter on the downstream
evolution of Eλk /3 is shown in Fig. 6, where the solid line
is an interpolation of 6.35-mm diameter for Rek = 177. In
this figure, the 6.35- and 7.62-mm-diameter roughness produce transient growth, 4.76- and 5.08-mm-diameter roughness
produce an initial decay followed by transient growth, and3.81-mm-diameter roughness produce pure decay. These findings indicate that both height and shape are equally importantin the receptivity of roughness-induced transient disturbances.
Tumin (2003) proposed a multimode decomposition
method that addresses the receptivity problem for spatiallygrowing realistic perturbations. Quantification of the initial
disturbance profile for comparison with this theory requires
the measurement of all three disturbance velocity components. Ergin and White (2005) took the first step in obtaining the complete velocity field by measuring the spanwise
and streamwise velocity components. At present, spanwise
velocity measurements are too limited for comparison withnumerical works. Continuing efforts are focused on obtaining
extensive streamwise and spanwise velocity measurements sothat a definitive comparison can be made. Multicomponent
velocity measurements are crucial to an improved understanding of transient disturbances, because of their relevance tothe streamwise oriented steady vortical structures that initiate transient growth.
DIRECT NUMERICAL SIMULATIONSFischer and Choudhari (2004) have carried out numericalsimulations for various roughness configurations that enabledetailed comparisons with the measurements of White and Ergin (2003). The unsteady incompressible Navier–Stokes equations were integrated in time by using the spectral element
discretization in space and a third-order, operator-splitting
formulation in time. After initial simulations with a full wavelength domain to confirm the spanwise symmetry of the flow
field, the remaining simulations were performed using half the
wavelength along the spanwise direction. Grid convergence ofthe computational results was confirmed by varying the orderof the discretization polynomial within each element.
For Rek = 119, Fischer and Choudhari (2004) examinedthe downstream evolution of the relevant spanwise harmonics
of the disturbance field. The agreement between the experimentally measured and numerically computed modal energies
can be observed by comparing Fig. 7 with Fig. 4. The initial
rapid decay of the λk ; the transient growth of λk , λk /3, andλk /4; and the monotonic decay of λk /2 are all captured in thesimulations. In Fig. 7, the numerically determined streamwise
locations where the fundamental mode, λk reaches a minimum, where λk /3, and λk /4 modes reach their respective peak
amplitudes, and where the downstream decay rate of the λk /2mode changes are almost exactly equal to their experimentallymeasured counterparts. Minor differences are observed in the
far wake region of the fundamental mode. The modal energy
computed using DNS overestimates the corresponding modal
energy measured in the experiments (modal energies can beobtained from the experimental data by integrating Eλ over
one unit of inverse wavelength). Numerical results for otherroughness heights are consistent, in general, with the observed
nonlinear scaling of disturbance amplitudes and the approximate range of Rek over which this scaling has been observed.
Figure 8 compares the downstream evolution of steady
disturbance energy obtained by experiments and DNS for
Rek = 119. The disturbance energy at each x location iscomputed by integrating the spanwise variance of the steadyvelocity disturbance from the wall to the freestream. This
figure shows good agreement between the approaches withboth indicating an initial rapid decay region, a slow but clear
growth region and finally a weak decay region in the x direction.
One of the advantages of DNS is the readily available pressure field, which cannot be easily measured in the experiments.
Figure 9 shows the salient features of the near-wall flow field in
the near vicinity of roughness element for Rek = 119. The bottom half of the figure displays the surface pressure contours.
One observes peak pressures along the centerline just aheadof the roughness element, lowest pressure above the roughness(not shown in the figure), and a monotonic pressure recovery
in the wake region, all in agreement with the previous works.
The upper half depicts the distribution of streamwise velocity and the streamlines at a small fixed height just above thesurface. The top half of the figure clearly highlights the two
separate regions of flow reversal - a primary region behind theroughness element and a smaller region near the centerline
just in front of the element.
THEORETICAL ANALYSIS
A theoretical analysis of perturbations generated by aroughness element was presented in Tumin and Reshotko(2004) who considered the linearized Navier–Stokes equations
and linearized boundary conditions. The method was basedon an expansion of the solution into the normal modes of thecontinuous spectrum and amplitudes were calculated using thebiorthogonal eigenfunction system (Tumin 2003). The linear
approach is limited to small roughness elements with heights
less than the viscous sublayer thickness. In the present experiments and computations, the roughness height exceeds thisby a significant margin. As follows from the high Reynolds
number asymptotic theory (Smith et al. 1977), the governing equations are linear in the boundary layer’s main deck,
but nonlinear in the vicinity of the wall. The experimental
and computational data demonstrate that the velocity perturbation amplitudes scale with Rek (i.e., k 2 ), illustrating thenonlinear character of the receptivity problem.
In order to assess the difference between the linear andnonlinear receptivity mechanisms, we compare the numericalresults for Rek = 119 based on the linear theory (Tumin andReshotko 2004) with DNS results and optimal perturbations.
Figures 10 and 11 show a comparison of the wall-normal andthe spanwise velocity components of the fundamental mode
at x = 335 mm. In both figures, the v and w profiles computed using each approach are normalized by the same factor
such that the near-wall maximum w velocity computed using
each method is unity. One can see that the velocity perturbations computed using linear receptivity theory are localized
closer to the wall than the DNS perturbations, and the DNS
perturbations are closer to the wall in comparison with theconventional optimal perturbation.
In spite of the nonlinear character of the receptivity phenomenon, the downstream perturbations are small enough,such that their dynamics in the transient growth region canbe described within the scope of the linearized Navier–Stokes
equations. The observed transient growth effect differs fromthe theoretical prediction based on the optimal perturbation
concept. To assess the difference between the optimal velocity
perturbations and the perturbations generated by the roughness array, we have calculated the optimal profiles corresponding to the maximum energy growth for the λk component overthe limited range of x = 335 mm to x = 700 mm. Figs. 10 and
11 confirm that the optimal perturbations are spread farther
away from the wall than the DNS profiles for the particularroughness configuration examined herein.
CONCLUSIONS
The recent developments in transient growth research obtained using experimental, theoretical and computational approaches are presented. Experimental results show that disturbances generated by surface roughness are non-optimal andthe DNS simulations are in satisfactory agreement with themeasured data. Non-optimality is attributed to the fact thatthe disturbance generated by the roughness element involves
a streamwise-oriented vortex that is closer to the surface than
the streamwise vortex predicted by optimal disturbance calculations. The experiments and simulations also establish thatthe disturbance velocity scales approximately with the squareof the roughness height and that the roughness shape also has
an important influence on the receptivity of roughness-induced
disturbances. The dependence on roughness height shows thatthe receptivity mechanism is nonlinear, which implies thatlinear receptivity models are not appropriate for the roughness amplitudes considered here. Despite the fact that thereceptivity is nonlinear, the downstream perturbations in thetransient growth region are small and, hence, their evolution
should be well approximated by the linearized Navier–Stokes
equations. Consequently, the multimode decomposition technique (Tumin 2003) can be applied to analyze the DNS andthe experimental data, providing a quantitative description of
the perturbations as a sum of the continuous spectrum normal
modes. This technique requires the measurement of the complete velocity field in the near wake. Current experimental
efforts are focused on supplying this data by measuring the w
and u velocities and estimating the v velocity using continuity.
This data, in conjunction with the biorthogonal decomposition technique, will lead to receptivity models for non-optimal
initial disturbances which will be useful for predicting transition in situations with significant roughness-induced transient
growth.
ACKNOWLEDGMENTS
The authors wish to thank Prof. Edward White for many
valuable comments and suggestions. FGE acknowledges the
Air Force Office of Scientific Research for its support throughgrant number F49620-02-1-0058 and the Case School of Engineering for its support through a Case Prime Fellowship.
Computational resources for the DNS were provided both by
NASA and the Department of Energy. AT acknowledges the
Air Force Office of Scientific Research for its support throughgrant number F33615-98-D-3601.

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Smith, F. T., R. I. Sykes, and P. W. M. Brighton (1977).
A two-dimensional boundary layer encountering a threedimensional hump. J. Fluid. Mech. 83, 163–176.
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REFERENCES
Andersson, P., M. Berggren, and D. S. Henningson (1999).
Optimal disturbances and bypass transition in boundary
layers. Phys. Fluids 11 (1), 134–150.
Butler, K. and B. Farrell (1993). Optimal perturbations andstreak spacing in wall-bounded turbulent shear flow. Phys.
Fluids A 5 (3), 774–777.
Chernyshenko, S. I. and M. F. Baig (2005). Streaks and vortices in near-wall turbulence. Phil. Trans. Royal Soc. A.
To appear.
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Farrell, B. F. (1988). Optimal excitation of perturbations inviscous shear flow. Phys. Fluids 31, 2093.

Roughness Arrayy

Measurement Planexz
λkFischer, P. and M. Choudhari (2004). Numerical simulation ofroughness-induced transient growth in a laminar boundary
layer. AIAA Paper 2004-2539.
Fransson, J. H. M., L. Brandt, A. Talamelli, and C. Cossu(2004). Experimental and theoretical investigation of thenon-modal growth of steady streaks in a flat plate boundary layer. Phys. Fluids 16 (10), 3627–3638.
Landahl, M. T. (1980). A note on an algebraic instability ofinviscid parallel shear flows. J. Fluid Mech. 98, 243–251.

xk

Figure 1: Schematic view of the flat plate model and roughness
array.

0.0085
4

Rek = 36
55
87
125
143
195

0.006
Eλk /3

3
η
2

0.004

0.002

1

0

0
-0.5

-0.167

0

0.167

300

0.5

400

500

600

700

x [mm]

z/λk
Figure 2: Experimentally measured streamwise velocity contours for x = 330 mm, Rek = 202.

Figure 5: Transient growth of disturbance kinetic energy contained in λk /3 for various roughness heights. d = 6.35 mm.

0.006

5

d=6.35 mm
3.81
4.76
5.08
7.62

4
0.004
Eλk /3

3
η
2

0.002

1
0

0
-0.5

-0.167

0

0.167

300

0.5

400

500

600

700

x [mm]

z/λk
Figure 3: Numerically calculated streamwise velocity contours
for x = 330 mm, Rek = 202.

Figure 6: Evolution of disturbance kinetic energy containedin λk /3 for various roughness diameters. Rek = 177.

0.003

Eλ

0.003

Modal Energy

λk
λk /2
λk /3
λk /4

0.002

0.001

0

λk
λk /2
λk /3
λk /4

0.002

0.001

0
300

400

500

600

700

x [mm]
Figure 4: Streamwise evolution of Eλ measured by White and
Ergin (2003) for Rek = 119.

300

400

500

600

700

x [mm]
Figure 7: Streamwise evolution of modal energies computed
by Fischer and Choudhari (2004) for Rek = 119.

0.005
DNS
Experiment

Erms

0.004
0.003
0.002
0.001
0
300

400

500

600

700

x [mm]
Figure 8: Downstream evolution
steady spanwise rms disR ∞ ′ ofturbance energy, Erms =Urms 2 dη for Rek = 119.
0

10
DNS
8

LinearOptimal

6
η
4
2
0
-2

-1

0

1

w

Figure 9: Flow features in the vicinity of a roughness element:
surface pressure distribution (lower half); streamwise shear
stress and streamlines (upper half).
10
DNS
8

LinearOptimal

6
η
4
2
0
-1

-0.5

0

v
Figure 10: Comparison of wall-normal velocity profiles for x =335 mm, Rek = 119.

Figure 11: Comparison of spanwise velocity profiles for x =335 mm, Rek = 119.