In this work, the Reduced Navier Stokes (RNS) are numerically integrated, and used to calculate nonlinear finite amplitude streaks. These structures are interesting since they can have a stabilizing effect and delay the transition to the turbulent regime. RNS formulation is also used to compute the family of nonlinear intrinsic streaks that emerge from the leading edge in absence of any external perturbation. Finally, this formulation is

generalized to include the possibility of having a curved bottom wal

Martel Escobar, Carlos

MartÍn Bautista, Juan A.

In this work, the Reduced Navier Stokes (RNS) are numerically integrated, and used to calculate nonlinear finite amplitude streaks. These structures are interesting since they can have a stabilizing effect and delay the transition to the turbulent regime. RNS formulation is also used to compute the family of nonlinear intrinsic streaks that emerge from the leading edge in absence of any external perturbation. Finally, this formulation is generalized to include the possibility of having a curved bottom wal

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Computation of nonlinear streaky structures in boundary layer flow

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YIC2013
Second ECCOMAS Young Investigators Conference
2–6 September 2013, Bordeaux, France
Computation of nonlinear streaky structures in boundary layer flow
J. A. Martin a,∗, C. Martel a
a ETSI Aerona´uticos, Universidad Polite´cnica de Madrid
Pza Cardenal Cisneros 3, 28040, Madrid, Spain.
∗juanangel.martin@upm.es
Abstract. In this work, the Reduced Navier Stokes (RNS) are numerically integrated, and used to calculate
nonlinear finite amplitude streaks. These structures are interesting since they can have a stabilizing effect and delay
the transition to the turbulent regime. RNS formulation is also used to compute the family of nonlinear intrinsic
streaks that emerge from the leading edge in absence of any external perturbation. Finally, this formulation is
generalized to include the possibility of having a curved bottom wall
——————————————————————————–
Keywords: streaks; nonlinear; boundary layer; intrinsic
1 INTRODUCTION
Streaks are three dimensional boundary layer flow structures that take the form of spanwise thin and streamwise
elongated regions of high and low speed flow that alternate in the spanwise direction. The resulting streamwise
velocity profile exhibits a strong modulation in the spanwise direction, with a characteristic scale of the order of the
boundary layer width. The slow transversal (wall-normal and spanwise) motion points downwards in the high speed
region and upwards in the low speed region (see Figure 1).
This work is focused on the streaks that are forced from inside the boundary layer, with no free stream perturbations,
by means of a spanwise periodic array of small cylindrical roughness elements attached to the plate near the leading
edge. (see, e.g., [1, 2, 4]), and they are interesting for their potential to extend downstream the laminar regime of
the flow, moving further downstream the location of the transition point. The stability of the boundary layer with
finite amplitude streaky distortions has been analyzed in [3, 4, 7], and experimentally tested [1, 2], detecting that
the increase of the amplitude of the streaks can have an stabilizing effect in the Blasius boundary layer, delaying the
onset of turbulence. The linear inviscid stability depends on the amplitude of the streak. In [5], it was found that
there is a maximum streak amplitude (∼ 26% of the free stream velocity) beyond which transition is promoted in
the boundary layer.
Nevertheless, some recent experimental results [6] have reported the possibility of inducing stable, robust steady
streaks with amplitude significantly above the ’critical one’ (∼ 26%), without having any secondary instability
acting on them. So, the streak stability characteristics are definitely far from clear, and that they probably depend
not only on the magnitude of its spanwise modulation but also on how the streaks are generated. Thus, there is a
clear motivation to compute high amplitude streaks.
2 RNS FORMULATION
High intensity streaks are nonlinear, and are typically computed using either direct numerical simulation (DNS)
or nonlinear parabolized stability equations (PSE). The DNS for the streak computations is very CPU costly since
this is a three dimensional problem that, in order to correctly reproduce the boundary layer conditions, has to be
performed at a very high Reynolds number [3, 4]. On the other hand, the nonlinear PSE has strong consistency
problems that require the addition of extra stabilization terms, and it simply fails to converge when the amplitude of
2 J.A. Martin et al. | Young Investigators Conference 2013
Figure 1: Sketch of the development of a spanwise
periodic array of streaks on a flat plate boundary
layer, with the asymptotic scaling for Re ≫ 1 in-
dicated. HS (LS) stands for high (low) streamwise
velocity.
1 2 3 4 5
0
0.1
0.2
0.3
0.4
0.5
B
C
D
E
X
A
s
 
 
Figure 2: Nonlinear streamwise streak amplitude
evolution, for different initial amplitudes: RNS
(blue line), PSE from [7] (red dashed line), and DNS
from [3, 4] (black line).
the streak is not small (see [7]). So, in this work, a much more efficient way is used to compute nonlinear streaks,
the reduced Navier-Stokes (RNS) equations.
A flat plate boundary layer at zero angle of incidence with a spanwise periodic array of counter-rotating steady
streaks developing in the streamwise direction is considered, see Figure 1. The velocities are made nondimensional
with the reference velocity the free stream flow, U∞, the spatial scales with a characteristic length L, and the
resulting Reynolds number is defined in the usual form, Re = U∞L/ν, where ν is the kinematic viscosity.
The RNS are obtained from the full 3D steady incompressible Navier Stokes equations in the limit of large Reynolds
number. The asymptotic structure of the streaks for Re ≫ 1 exhibits (Figure 1) a slow spatial dependence in the
streamwise direction, and two short spatial scales, in the normal and spanwise direction. This scaling is similar to
that of the standard 3D Boundary Layer equations (BL) [11] but with two short scales instead of just one.
The appropriate expansions for the flow variables are of the form
xˆ = x, (yˆ, zˆ) = (y, z) /
√
Re, uˆ = u+ . . . , (vˆ, wˆ) = ((v, w) + . . . )/
√
Re, pˆ = p0 + p1/Re+ . . . ,
which, once inserted into the Navier Stokes equations yields, at first order,
p0y = 0 and p0z = 0.
The pressure is then p0 = p0(x), given by the prescribed inviscid pressure at the upper edge of the boundary layer
(as in the standard BL formulation). And, at next order, the full set of RNS is obtained:
ux + vy + wz = 0,
uux + vuy + wuz = −p0x + uyy + uzz,
uvx + vvy + wvz = −p1y + vyy + vzz,
uwx + vwy + wwz = −p1z + wyy + wzz .
(1)
The RNS are a truly parabolic system in x that can be solved by marching techniques [8, 9]. In contrast with the
standard 3D BL formulation, now the second order y and z momentum equations are required to complete the
system, and the pressure correction term, p1, has to be computed in order to solve problem.
The RNS equations are well known [9, 10]. They have been previously used to compute high Re number micro
channel and micro tube flow [12], the nonlinear development of Go¨rtler vortices [13], the transient growth of small
amplitude streaks [14, 15, 16], or the streak excitation by vortical structures on the free stream, both in steady and
unsteady conditions. [17, 18],
The full 3D RNS system (1) is integrated to compute the nonlinear spatial evolution of streaks on a flat plate
boundary layer. The appropriate boundary conditions for this case are periodicity in z (i.e., a spanwise periodic array
of streaks), together with no slip at the bottom wall, and, at the upper edge of the boundary layer, all components of
v must vanish at y →∞ except for its mean value < v >z computed as part of the solution [16].
The RNS formulation allows to perform 3D streaky BL computations with much less CPU cost than previous DNS
computations[3], and avoid the numerical problems of the PSE simulations [7].
J.A. Martin et al. | Young Investigators Conference 2013 3
3 STREAK SIMULATIONS
In Figure 2 are presented the RNS results for the simulation of fully nonlinear streaks. The streamwise evolution of
the amplitude of the streak,
As(x) = (max
y,z
(u)−min
y,z
(u))/2,
is plotted for the same four streaks that were computed using DNS in [3], and PSE in [7]. The integration is started
at the station x0 = 0.4, and the initial profile data is taken from [7]. The results show that the agreement with the
DNS and PSE data is quite good: the difference between both results grows with the amplitude of the streak, but
never exceeds 3%. Note that the RNS results are always closer to the DNS than the PSE, and that the PSE was
simply not able to complete the computation of the largest streak (labeled E).
An interesting way to visualize the tree dimensional streak flow dynamics is to follow the downstream evolution
of the particle trajectories departing from a horizontal line located at the beginning of the computational box:
x = 0.4, y = 3. The resulting surfaces of particle trajectories are shown in Figure 3. The streaks induce a
counter-rotating motion in the (z, y) plane that gives these trajectory surfaces a characteristic mushroom-like shape
in the transversal section; very evident in the case of a high intensity streak (As0 = 0.4) depicted in the right plot of
Figure 3. Analyzing these particle trajectories could be helpful to understand a strong quantitative discrepancy be-
tween the streak spanwise period measured in some experiments using smoke visualizations [19], and that predicted
by the linear optimal theory [5, 15].
Figure 3: Surfaces of particle trajectories departing from the line x = 0.4, y = 3, for the streaks with initial
amplitudes As0 = 0.1 (left) and As0 = 0.4 (right). Several representative particle trajectories are also plotted with
blue lines.
4 INTRINSIC STREAKS
In [14] the flow near the leading edge of a flat plate boundary layer is analyzed using the linearized problem around
the Blasius solution. It is found that there is just one single streaky mode (periodic in the spanwise direction) that
grows downstream from the leading edge. The existence of this growing mode indicates that there is a one param-
eter family of 3D steady streak solutions that emerge from the leading edge of the boundary layer. The resulting
expansion of the solution for the growing mode of the velocity profile near the leading edge can be expressed then
as
u = UBlasius + ax
1−λU˜ (η) cos (z) + · · · ,
v = VBlasius + ax
1/2−λV˜ (η) cos (z) + · · · , (2)
w = −ax−λ
(
h (η)−√xV˜ (η) + · · ·
)
sin (z) ,
where λ = 0.7865 . . ., a is a free parameter, and the profiles for h(η), U˜(η), and V˜ (η) are obtained in [14] and [16].
The near leading edge expressions (2) obtained are used here as initial conditions for the RNS equations in order
to extend downstream the computation this family of intrinsic streaks (intrinsic because they appear in complete
absence of any free stream perturbations) using the RNS formulation.
The evolution of the streamwise and spanwise velocity component is represented in Figure 4 for different values
of the a parameter. As it grows, nonlinear effects appear and the evolution of the maximum moves away from the
linear theory predictions.
4 J.A. Martin et al. | Young Investigators Conference 2013
10−8 10−6 10−4 10−2 100
10−3
10−2
10−1
100
X
m
ax
|U
−
U
b
|
a
theoretical growth
10−8 10−6 10−4 10−2 100
100
105
X
m
ax
|W
|
linear decay
10−2
102
a
Figure 4: Downstream evolution of maximum deviation from Blasius profile of the streamwise velocity (left) and the
maximum of the spanwise velocity component (right). Solid lines: computations for a = 0.125, 0.25, 0.5 and 0.6,
arrow indicates increasing values of a. Dashed line: asymptotic behavior of the solution near leading edge.
5 CONCLUSIONS
RNS formulation has been used to describe the downstream evolution of a spanwise periodic array of fully nonlinear
streaks in a flat plate boundary layer, and the intrinsic nonlinear streaks that emerge from its leading edge, with
a much lower computational cost than DNS, and also more robust than nonlinear PSE. When the amplitude of
the streak increases, the nonlinear terms come into play and the flow configuration changes substantially, and the
transversal motion is important to understand the flow pattern.
The RNS formulation can be used in other fluid configurations provided that, at high Reynolds number, they exhibit
two short and one long length scale. For example, the development of cross-flow vortices in a swept wing [20].
ACKNOWLEDGEMENT
The authors thank Marı´a Higuera, Shervin Bagheri and Carlo Cossu for allowing us to access their results and G
Martı´n for his help with the 3D sketches. This work has been partially supported by the Spanish Ministerio de
Ciencia e Innovacio´n (grant TRA2010-18054) and by the Universidad Polite´cnica de Madrid (grant GI 11011039).
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Computation of nonlinear streaky structures in boundary layer flow

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