shear layer stability analysis in later boundary layer transition … · 2018-10-26 · american...

http://www.uta.edu/math/preprint/

Technical Report 2012-09

Shear Layer Stability Analysis in

Later Boundary Layer Transition and MVG Controlled Flow

Yonghua Yan Chaoqun Liu

American Institute of Aeronautics and Astronautics

1

Shear Layer Stability Analysis in Later Boundary

Layer Transition and MVG controlled flow

Yonghua Yan1 and Chaoqun Liu2

University of Texas at Arlington, Arlington, Texas, 76019

Abstract A 1-D and 2-D cylindrical flow stability is investigated in this paper. A typical velocity profile in a shear layer with infection point is studied first. The fluid motion is decomposed

to two parts: the rotational part and shear part. Linear stability analysis shows the

rotational part (like rigid body) is stable but the shear part is unstable. Of course, the shear

layer with inflection point is unstable as well. Following this analysis, a velocity profile taken

from the later boundary layer transition and another one from flow field generated by the

micro vortex generator are taken as the base flow. A general mechanism of vortex ring

formation has been observed, that an olive-shaped momentum deficit zone is formed by

vortex ejection and the momentum deficit forms a strong shear layer. Finally, vortex rings

are formed around the olive-shaped low speed zone which generates a shear layer.

Keywords: Shear Layer, Stability, Vortex Ring, Flow Transition

Nomenclature

Re = Reynolds number

L = Length,Distance

Vv

= Velocity vector

p = Pressure

∞ = Far field value

2λ = const representing vortex tube surface

inδ = inflow displacement thickness

I. Introduction

The transition process from laminar to turbulent flow in boundary layer is a basic scientific problem in modern

fluid mechanics and has been the subject of study for over a century. Many different concepts for the explanation of

the mechanisms involved have been developed based on numerous experimental, theoretical, and numerical

investigations. After over a century study on the turbulence, the linear and early weakly non-linear stages of flow

transition are pretty well understood. However, for late non-linear transition stages, there are still many questions

remaining for research1-5.

We conducted a large grid DNS to study the formation of the ring-like vortex6. A sixth order compact scheme is

used for spatial discretization in streamwise direction and wall normal directions7. In the spanwise direction where

periodic conditions are applied, the pseudo-spectral method is used. The governing equations are solved explicitly

by a 3rd order TVD Runge-Kutta scheme. The adiabatic and the non-slipping conditions are enforced at the wall boundary on the flat plate. On the far field and the outflow boundaries, the non-reflecting boundary conditions 8 are

applied.

The computational domain is displayed in figure 1. The grid level is 1920×128×241 representing the number

of grids in streamwise (x), spanwise (y), and wall normal (z) directions. The parallel computation is accomplished

through the Message Passing Interface (MPI) together with domain decomposition in the streamwise direction. The

computational domain is partitioned into N equal-sized subdomains along the streamwise direction. N is the number

of processors used in the parallel computation. The flow parameters and other information can be found in Ref 6.

1 PhD student 2 Professor and AIAA associate fellow.


2

Figure 1. Computational domain

To investigate the complicated flow structure, we use the 2λ criterion (Jeong & Hussain, 1995) for visualization.

Figure 2. Multiple ring formation at early stage

Figure 3. Vortex structure shown by

2λ for the transition process

From Fig 2 and Fig 3 we can see that the ring-like vortices shown by 2λ method play a key role in the flow

breakdown and transition. From our previous studies6, we found that they generate rapid downward jets which

induce the positive spike and bring the high energy to the boundary layer and cause the upward jets to mix the

boundary layer. In other words, the ring-like vortex formation and development is a key topic for turbulence study.


3

It sounds clear that there is no turbulence without ring-like vortices. Another very interesting topic is the evolution

of these ring-like vortices, we found that once they are formed they will never break down when they propagating to

the downstream. Ring-like vortex structure in the transition process is a very robust structure, we believe it's the key

issue of the study.

Micro vortex generators (VG) are a kind of low-profile passive control device designed for the boundary layer

control. In contrast to the conventional VG (widely used in aviation applications and with height (h) of the order of the boundary-layer (δ)), micro VG has a height approximately 20-40% (more or less) of the boundary layer. Among

these micro VGs, mircoramp vortex generators (MVG) are given special interest by engineers because of their

structural robustness. MVG generate a pair of streamwise vortex, which remains in the in the boundary layer for

relatively long distance, and the corresponding “down-wash” effect will bring momentum exchange to the boundary

layer which makes it less liable to be separated. During such process, a specific phenomenon called “momentum

deficit”, i.e., a cylindrical region consisted of low speed flows, will be formed after the MVG9. In Lin’s review10 on

the low-profile vortex generator, it was mentioned that a device like MVG could alleviate the flow distortion in

compact ducts to some extent and control boundary layer separation due to the adverse pressure gradients. Similar

comments were made in the review by Ashill et al11. The formal and systematic studies about the micro VGs

including micro ramp VG can be found in the paper of Anderson et al12.

In our previous study13, we try to understand the mechanism of the flow structure expecially the vortex structure

behind the MVG. Numerical simulations are made on ramp flow with MVG control. The geometry of the MVG and the computational domain is displayed in Fig 4 and Fig 5.The flow field around the MVG and surrounding areas has

been studied in details. As a result, the hairpin vortex was formed and then traveled downstream.

Figure 4. The sketch of MVG

MVG RegionFore-regionRamp RegionGrid

transition

Grid

transition

h

5.618.63875

1.875

Figure 5. The schematic of the half grid system for MVG case

To reveal the coherent structure of the flow, the iso-surface of λ2 scalar field is given in Fig. 6. It is very clear that there is a chain of vortex rings, starting from behind of the trailing-edge of MVG. The rings are generated

almost erectly at first and then be continuously distorted and enlarged while propagating downstream.


4

(a) global view

(b) close-up view behind the MVG

Figure 6. Vortex rings shown by iso-surface of λλλλ2

In our previous study13, it has been proved that the ring structure which is similarly found in the transition study

plays a very important role in flow control. What is also very interesting is those vortex structures generated by the

MVG are very robust, they never break down during their evolution. Two different high order numerical methods (DNS with compact scheme in transition problem and LES with

WENO scheme in flow control by MVG) are used for the two problems mentioned above, but we got the similar

ring-like structures and both played the critical role.

What is a coincidence is that we found low speed zone or momentum deficit region inside both ring-like

structures mentioned above. Fig 7(a) shows that vortex rings are formed around the olive-shaped low speed zone in

the flow transition simulation, and Fig 7(b) shows the similar 3D low speed cylindrical zone inside the rings

generated by MVG.

(a)


5

(b)

Figure 7. (a) 3-D momentum deficit and Ring-like vortices in transition problem, (b) The iso-surface of the streamwise velocity behind the MVG

The existence of olive-shaped or cylindrical low speed zone means a typical 3D sheer layer will be formed

around the boundary of low speed zone. The instability of sheer layer will roll up the vortex lines and thus produce

vortices. Ring-like vortices will be generated as result of the instable of sheer layer around the cylindrical boundary.

As a conclusion, we consider the ring-like structure is a fundamental phenomenon in most of the flow pattern and its

mechanism, especially the stability of the sheer layer around the low speed zone and the robustness of the structure,

should be studied in detail.

II 1-D Shear Layer Stability Analysis 2.1 Velocity Decomposition

Fluid motion can be decomposed as two parts: rotational part like rigid body and shear part when we ignore the

translation part:

)(2

1)(

2

1)(

2

1

)()(

TTT VVVVVVV

VXdVd

VdXVXXV

rrvrrrrrr

rrr

rrrrrr

∇−∇+=∇−∇+∇+∇=∇

∇•=

+=∆+

ε

(1)

The second part is a rotational part and the first part is the shear.

2.2 Velocity decomposition for a typical shear layer

A typical 1-D shear layer can be described by following functions:

)tanh(1 byaU =r

(2)

Which can be divided by a rotational part and a shear part (see Figure 8 and 9)


6

u1

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-2 -1 0 1 2

u1

u2

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

-2 -1 0 1 2

u2

u1-u2

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

-1 -0.5 0 0.5 1

u1-u2

(a) Shear layer with inflection point (u1) (b) Rotational part (u2) (c) Shear part (u1-u2)

Figure 8: Shear layer velocity decomposition

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-3 -2 -1 0 1 2 3

u1

u2

u1-u2

Figure 9 Shear layer and its shear and rotational parts

2.3 . Derivation of Linear Stability Equation

=⋅∇

∇+−∇=∇⋅+∂

∂

0

Re

1 2

V

VpVVt

V

(3)

Equation (1) denotes the incompressible and non-dimensional Navier-Stokes equation in which, ( , , )V u v w=

is the velocity vector. Considering that

),,(')(),,( 0 tyxqyqtyxq += (4)

where q can be specified as ( , , , )u v w p , 0 0 0 0 0( , , , )q u v w P= which indicates the value of mean flow, and

'q denotes the corresponding linear perturbation. By eliminating the second order perturbation terms, the governing

equation for small perturbations can be written as,

0'

Re/'')'(')(' 2

00

=⋅∇

∇=∇+⋅∇+∇⋅+∂

∂

V

VpVVVVt

V

(5)

As a first step, a localized 2-D incompressible temporal stability for shear layer is conducted. Actually, it relates to the distance among two neighboring vortices in the central streamwise plane. Assume the normal mode is

u


7

( )

( )

ˆ ( ) . .

ˆ ( ) . .

i x z t

i x z t

V V y e c c

p p y e c c

α β ω

α β ω

+ −

+ −

= +

= +, (6)

where the parameter α is given, which is real and set according to the averaged distance between the new generated rings, and c should be a complex number. By plugging in Equation (3), Equation (2) can be rewritten as,

0ˆ)ˆˆ(

ˆReˆ

)ˆRe(ˆ

ˆReˆ)Re(ˆ0

=++

=

=

+=

vDwui

piwL

pDvL

pivDuuL

βα

β

α

(7)

where )}Re()({ 0

222 ωαβα −−+−= uiDL , and dr

dD =

Considering in 2D case (without w), and by eliminating ˆ ˆ,u p , we can obtain the standard O-S equation on v̂ ,

2 2 2 2 2 2

0 0ˆ ˆ( ) Re[( )( ) ] 0D v i U c D D U vα α α− − − − − = (8)

Equation (5) is about v̂ , but we need to get the value of c. The value of c determines the property of stability of the

equation. Let r ic c ic= + , if 0ic > , then the disturbance will continuously grow and the flow would be instable.

Otherwise, the flow would be stable.

2.4 Stability Analysis to the Given Velocity Profile

If there is no disturbance at the boundary and it will be free stream outside the domain (a, b), then we have the

corresponding boundary condition for function v̂ as ˆ ˆ( ) ( ) 0v a v b= = and ˆ ˆ( ) ( ) 0Dv a Dv b= = . The second order

central difference scheme (Equation (6)) is used to derive the finite different equation from Equation (4),

+−+−=

+−=

−−++

−+

2

2112

4

2

11

2

/)464(

/)2(

hD

hD

nnnnn

nnn

ψψψψψψ

ψψψψ (9)

Apply (6) to Equation (5) we can get the generalized eigenvalue problem:

0=+ ϕϕ BcA (10) Where A (symmetric pentadiagonal matrix) and B (symmetric tridiagonal matrix) are the coefficients’ matrix

and the vector ϕ denotes the values of v̂ at different position. c becomes the generalized eigenvalue of Equation

(10).

To simplify the process and also to reveal the mechanism of the stability to the velocity profile given in Fig 1a

(the circle part), an analysis on the corresponding fitting curve is made (see Fig 2, u1).

By solving the general eigenvalue problem (7), we can get the physical solution of the frequency c,

whose imaginary part ic is about 0.76 for our case. The positive value means this kind of flow is

unstable. In Fig. 10, v1 shows the corresponding shape of the eigenvector function, v̂ .


8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

v1

v2

Figure 10. Corresponding shape of the eigenvector functions for u1 and u1-u2 velocity profile.

By given the same flux as u1, we can obtain a velocity profile with linear distribution (u2). The physical

significance of the profile u2 represents the rotation part of the velocity distribution given by u1. Thus, the

difference velocity part, given by u1-u2 in Fig 9, has the meaning of corresponding deformation.

The same linear stability analysis process is applied on both the rotation part and the difference velocity

profile, and we can find that the profile of u2 gives a stable result while the difference velocity profile is proved to

be unstable with ic as 0.18.

To avoid the influence of the existence of the 2 extra reflection points on the difference velocity profile,

stability analysis are also applied on all of the three velocity curves by shrinking the domain of y (the height) to [-

0.7,0.7]. However, ic for u1 is 0.36 and ic for u1-u2 is 0.26, and the curve for u2 is still stable which means the

results do not change.

In the practical fluid flow, especially for the flow pattern which we can find inside the vortex ring, u2

represents the rigid rotation of the vortex tube. However, through stability analysis, we can come to the conclusion

that such kind of rigid rotation is unconditionally stable which makes the ring structure to be so robust.

III Stability Analysis for Late Boundary Layer Transition

The distributions of averaged streamwise-velocity are given in Fig. 11a along the normal grid lines at the center

plane. The streamwise positions of the line is x=491.1inδ . The dip of the line corresponds to the momentum deficit.

It can be seen clearly that there a high shear layer in the central plane which is located above the ring legs. The

second order derivative 22/ zU ∂∂ is calculated to demonstrate the existence of the inflection points. The existence

and correspondence of the inflection point at the upper is illustrated by two dashed lines intersecting the distribution

of the streamwise velocity and its second order derivative (Fig. 11b).

y


9

Figure 11a: The streamwise velocity distribution Figure 11b: The distribution of second order derivative

of corresponding streamwise velocity

It is obvious that the existence of the inflection points (e.g. z=5.2) in shear layers will cause the flow instability

and generates vortex rollers according to the shear layer instability theory in 2-D. So the mechanism for the vortex

ring generation is closely related to the shear layer instability. Loss of the stability of the shear layer will result in the

formation of the vortex rings for 3-D flow.

The second order central difference scheme is used to get the finite different equation from equation (3), then a

so called eigenvalue method is applied to get the value for c which should be a complex number. We can get the

physical solution of the frequency c , whose imaginary part ic is about 0.045 for our case. The positive value of ic

means the shear layer is unstable. Fig.12 shows the corresponding shape of the corresponding eigenvector function

)(yψ and Figure 6 shows that vortex rings are formed around the olive-shaped low speed zone which is a shear

layer.

Figure 12. Shape function


10

IV. Stability Analysis for Flow Field Generated by Micro Vortex Generator

The flow field generated by micro vortex generator has been studied as well. Time and spanwise averaged

streamwise-velocity given at Lfrom apex/h ≈ 3.3 is plotted in Fig. 13(a). The existence and correspondence of the inflection points at the upper and lower shear layers is clearly illustrated as follow (Figure 13b).

(a)

(b)

Figure 13 Inflection Points (surface for 3-D) (a) Averaged Streamwise Velocity at Lfrom apex/h ≈3.3 (b) ∂2w/∂y

2

Similar to the linear analysis did in above section, the second order central difference scheme is used and the

eigenvalue method is applied to get the value for c. According to the normal mode in Eq. (2), we can get the

physical solution of the frequency c, whose imaginary part ic is about 0.068 for our case. The positive value means

this kind of flow is unstable. Fig. 14 shows the corresponding shape of the eigenvector function, v̂ .


11

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 14: Eigenvector function of v̂

V. 2-D stability analisys

This part is still under development and we will report the progress in the AIAA 2013 Aerospace Science Meeting.

VI. Conclusion

Fluid motion can be decomposed as a shear part (symmetric) and rotational part (asymmetric). The rotational

part is always stable, but the shear part with inflection point is in general unstable. The momentum deficit is always

generated by vortex ejection and a consequent olive-shaped shear layer is generated by the momentum deficit. These

shear layers eventually generate ring-like vortices. This mechanism is believed universal for multiple vortex ring

formation in both late flow transition and flow field generated by micro vortex generator.

VII. Acknowledgments

The authors are grateful to Texas Advanced Computing Center (TACC) for providing computation hours.

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Mech.1991.23:495-537

[2]Boroduln V I, Gaponenko V R, Kachanov Y S, et al. Late-stage transition boundary-Layer structure: direct

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[3]Bake S, Meyer D, Rist U. Turbulence mechanism in Klebanoff transition:a quantitative comparison of

experiment and direct numerial simulation. J.Fluid Mech, 2002 , 459:217-243

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[5]Kachanov, Y.S. On a universal mechanism of turbulence production in wall shear flows. In: Notes on

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— Berlin: Springer, 2003, pp. 1–12.

[6]Liu, C. and Chen, L., “Study of Mechanism of Ring-Like Vortex Formation in Late Flow Transition,” AIAA Paper 2010-1456

[7]Lele, S.K. Compact finite difference schemes with spectral-like resolution. J. Comp. Phys. 1992, 103, p. 16.

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Comput. Phys. 1988, 77, pp. 439-471.

[9]Babinsky, H., Li, Y., and Ford, C.W.P., "Microramp Control of Supersonic Oblique Shock-Wave/Boundary-

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12

[10]Lin J. C. "Review of Research on Low-Profile Vortex Generators to Control Boundary-Layer Separation."

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shear layer stability analysis in later boundary layer transition … · 2018-10-26 · american...

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