# article .pdf

À propos / Télécharger Aperçu

**article.pdf**

Ce document au format PDF 1.4 a été généré par TeX output 2011.12.11:0744 / dvipdfmx (20090708), et a été envoyé sur fichier-pdf.fr le 11/12/2011 à 13:58, depuis l'adresse IP 78.233.x.x.
La présente page de téléchargement du fichier a été vue 1259 fois.

Taille du document: 4.3 Mo (29 pages).

Confidentialité: fichier public

### Aperçu du document

A study of unsteady boundary-layer separation

and vortex shedding

Clement DAUBRENET

December 11, 2011

Contents

1 Introduction

1.1 General context of the study . . . . . . . . . . . . . . . . . . . .

1.2 Theory of unsteady separation . . . . . . . . . . . . . . . . . . .

3

3

4

2 Problem formulation

2.1 General case . . . . . . . .

2.2 Channel 1 . . . . . . . . .

2.3 Channel 2 . . . . . . . . .

2.4 Solver . . . . . . . . . . .

2.5 Computational framework

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

5

5

6

7

8

9

3 Navier-Stokes solution

3.1 Steady separation, low Reynolds numbers, α = 0.50 .

3.2 Incipient separation and marginal separation theory

3.3 Unsteady separation, Re=2000, α = 0.50 . . . . . . .

3.4 Unsteady separation, Re=10000, α = 0.70 . . . . . .

3.5 Vortex shedding, Re=105 , α = 0.70 . . . . . . . . . .

3.6 Vortex shedding, Re=105 , α = 0.40 . . . . . . . . . .

3.7 Vortex shedding, Re=105 , α = 0.60 . . . . . . . . . .

3.8 Vortex shedding, Re=105 , α = 0.70 . . . . . . . . . .

3.9 Analysis of the results . . . . . . . . . . . . . . . . .

3.10 Strouhal number . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

10

10

11

12

14

16

18

19

20

21

22

4 Theory of turbulence modelling

4.1 The Reynolds equation . . . . . . . . . . . . . .

4.2 The Turbulent-viscosity hypothesis . . . . . . .

4.3 The energy cascade . . . . . . . . . . . . . . . .

4.4 The Turbulent Kinetic Energy (TKE) equation

4.5 The k − model . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

23

23

24

25

25

25

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

5 Influence of the constants on

5.1 The log-law . . . . . . . . .

5.2 Impact of a κ modification .

5.3 The Cµ coefficient . . . . .

separation

26

. . . . . . . . . . . . . . . . . . . . . 26

. . . . . . . . . . . . . . . . . . . . . 27

. . . . . . . . . . . . . . . . . . . . . 27

6 Conclusion

28

2

1

1.1

Introduction

General context of the study

Understand the mechanism of flow separation is crucial and applications in different fields are numerous. Sport is one of them. In golf, aerodynamics of the

club head is becoming a major preoccupation : “One area is the aerodynamics

of the head. There are no limits on drag coefficients, for example. Our competitors will, of course, say similar things but we have discovered that this area

is very complicated” says Dr.Alan Hockwell team leader of the Callaway R&D

department. The same is done in a myriad of different sports where speed or

distance have a real importance : tennis, swimming and windsurfing for example. In cycling, the research on helmets is growing fast : “Most people dont

realize that a non-aero helmet creates four times the drag of a non-aero wheel

set” says Dr.Mark Cote from the Massachussets Institute of Technology.

Figure 1: Subsonic aerodynamics in sport

Various others applications require an understanding of separation, unsteady

separation and vortex shedding in the industry. Aerospace is a classical example,

mainly for transonic and supersonic flows : in the study of winglets and airfoils

design but also in the compressor where stall must be avoid. Vortex shedding

measurement thanks to Fluidic Flowmeters is extensively used in wastewater,

mining, mineral processing, power, pulp and paper, chemical, and petrochemical

industries. It allows to know the different flowrates by determining the frequency

of shedded vortices. The industry of trucks is also implied because any fuel

saving is highly seeked due to the number trucks and miles involved.

Figure 2: Subsonic aerodynamics in industry

3

1.2

Theory of unsteady separation

The first person to define flow separation formally was Prandtl in 1904. He

stated that separation occurs when the wall shear stress reaches zero. This

definition only stands for incompressible steady separation. What happends

when the Reynolds number and the adverse pressure gradient make the flow

unsteady ? A generic theory has been developed over year. In 1975, the MooreRott-Sears (MRS) model defines unsteady separation as the onset of a finite-time

singularity occuring along a zero vorticity line. Three stages are indentified both

in time and in model complexity :

Figure 3: Schematic of the initial asymptotic structure of unsteady separation

The first stage is the non-interactive boundary-layer stage driven by the

boundary-layer equations. Van Dommelen showed in 1981 that this stage ends

with a singularity at ts . A sharp spike errupts from the wall. Going to the second stage implies a small-scale viscous-inviscid interaction (second tier in figure

1). In the last stage (normal pressure gradient), the vorticity leaves the boundary layer. This stage evolves towards vortex shedding and is still not clearly

characterized. Three different Reynolds number regimes can be identified, as

done in [2], for a vortex induced separation. The first regime is a low-Reynoldsnumber regime (Re = 103 ). It reveals a large-scale interaction only, that is to

say no spike (small-interaction) is visible. The streamlines stay globaly smooth

and streamwise pressure gradient as well. The second regime is the moderateReynolds-Nnmber regime (Re = 103 ). Both large-scale interaction and smallscale interaction occur. The streamwise pressure gradient disturbances lead to

a spike in the streamlines and at the minimum of the streamwise pressure gradient. The first indicator of it is a small disturbance in wall shear stress (the

streamlines begins to leave the wall in the normal direction). The last regime

is for very high Reynolds number (Re > 105 ). We can’t distinguish any clear

shift or disturbances in the streamwise pressure gradient (no large-scale interaction). However, the spike (small-scale interaction) is coming quickly, both in

streamlines and streamwise pressure gradient.

4

2

Problem formulation

2.1

General case

In this project, we focused on a simple geometry. A bidimensionnal channel

with a suction slot generating an adverse pressure gradient is considered :

Figure 4: Nondimensionalized channel with suction slot

We define the parameters of this study :

• The velocities vi for the inlet, vs for the suction, vo for the outlet

• The flow rates Qi for the inlet, Qs for the suction, Qo for the outlet

• The suction ratio α defined as the ratio of Qs over Qi

• The Reynolds number defined as the ratio of the velocity u over the viscosity of the flow ν

The analysis done in [1] leads to the following definition of the suction

velocity, for t > ti , which is gradually turned on :

√

b

vs (x) = α(t) √ exp(−bx2 )

π

Where :

(t − ti )3

]}αf

0.25

From t = 0 to t = ti , no adverse pressure gradient is applied and the boundary

layer develops. When the flow reaches steadyness, the suction is turned on at

ti = 1. In the following, the suction ratio is αf which is the non-time dependent

version of α.

α(t) = {1 − exp[−

5

2.2

Channel 1

The first geometry is a short channel, designed for the study of unsteady separation for simulations from t=0 to t=6. Thus, the non-dimensionalized length

of the channel is taken as L=4. As a result, the outlet will not influence the

results.

Figure 5: Channel for unsteady separation study

The mesh is chosen to have the best results in the wall area : fine in the x

direction with a contraction as y goes to zero.

Figure 6: Mesh for geometry 1. For visual clarity, this figure is not at scale

Here is, for different grids, the degree of independance solution reached at

t=2 :

(382x18) to (764x36)

(764x36) to (1528x64)

∆ CPU

15 mins

1 hour

Max pressure

4.2 %

3.9 %

Max streamfunction

3.7 %

2.6 %

We select the (764x36) mesh. Thus, the error is around 4 % for the pressure

and 2.5 % for the streamfunction with respect to the finest grid.

6

2.3

Channel 2

The second geometry is a long channel designed for vortex shedding study for

simulations from t=0 to t=35. Thus, the non-dimensionalized length of the

channel is taken as L=25. As a result, the outlet will not influence the results.

Figure 7: Channel for vortex shedding study

The mesh is chosen to have the best results in the suction area : fine in the

y direction with a contraction as x goes to 1.

Figure 8: Mesh for geometry 2. For visual clarity, this figure is not at scale

Here is, for different grids, the degree of independance solution reached at

t=2 :

(277x7) to (554x14)

(554x14) to (1108x28)

(1108x28) to (2216x56)

(277x7) to (2216x56)

∆ CPU

10 mins

2 hours

22 hours

24 hours

Mean pressure

6.4 %

6.3 %

4.0 %

14.4 %

Mean streamfunction

4.2 %

3.8 %

1.9 %

10 %

In the worst case, our error will be 14.4 % for the pressure and 10 % for the

streamfunction with respect to the finest grid.

7

2.4

Solver

OpenFoam provides several pre-built solvers. To perform this simulation, PISOFOAM (Pressure-Implicit with Splitting of Operators, Issa 1985) is used because

adapted for transient incompressible flows. Let us consider the Navier-Stokes

equation for incompressible flows :

∂u

+ (u · ∇) · u = ∆u − ∇p

∂t

∇(ρ · u) = 0

To solve it, we need an equation for the pressure. The common approach is

to derive an equation by taking the divergence of the momentum equation and

by substituting it in the continuity equation. We obtain :

∇(

X H(u)

1

H(u)

∇p) = ∇(

)=

S(

)f

ap

ap

ap

(1)

f

Where S is outward-pointing face area vector and f the mesh faces.

Then the PISO algorithm can be summed up as follows :

• 1 - Set the boundary conditions

• 2 - Solve the discretized momentum equation to compute an intermediate

velocity field

• 3 - Compute the mass fluxes at the cells faces

• 4 - Solve the pressure equation

• 5 - Correct the mass fluxes at the cell faces

• 6 - correct the velocities on the basis of the new pressure field

• 7 - Update the boundary conditions

• 8 - Repeat from 3 for the prescribed number of times

• 9 - Increase the time step and repeat from 1

8

2.5

Computational framework

We use different tools in this study. The solver PISOFOAM solves NavierStokes, while the OPENFOAM viewer, PARAVIEW, allows to plot the fields.

Some modifications in C++ to implement time-dependent boundary conditions

are required. Finally, Python is used for the grid independence study and to

sample pressure data before the Fast Fourier Transfort (FFT) analysis in MATLAB. Here is a shematic of the framework :

Figure 9: The framework. White : OPENFOAM evironment, blue : added

analysis tools

9

3

Navier-Stokes solution

We examine different Navier-Stokes solutions for several Reynolds number (Re)

and suction ratio (α). We try to identify some patterns of steady separation

as well as unsteady separation. Lastly, a frequency analysis of vortex shedding

will be performed.

3.1

Steady separation, low Reynolds numbers, α = 0.50

We show different steady solutions for a given adverse pressure gradient. As

the Reynolds number increases, the length separation bubble increases and its

center moves upstream. The same observation is provided in [3]. Notice the

zero-vorticity in red.

Figure 10: Re = 454, streamlines (left), vorticity countours(right)

Figure 11: Re = 500, streamlines (left), vorticity countours(right)

Figure 12: Re = 625, streamlines (left), vorticity countours(right)

10

3.2

Incipient separation and marginal separation theory

We compare the incipient separation results from our Navier-Stokes solutions

with marginal separation theory (Ruban 1981, 1982; Stewartson, Smith and

Kaups 1982). The results from Cassel in [1] are also considered. Recall that

the marginal separation theory predicts the incipient separation for this suction

ratio :

αi = αm + cRe2/5 Γs

(2)

where αm = 0.05167, c=2.33 and Γs = 2.28. We compute the wall shear

stress with a third order derivative of the streamwise velocity on the finest grid.

We consider unsteady separation when the wall shear stress reaches zero.

Figure 13: Marginal separation theory, Cassel’s results (green), results from

PISOFOAM (red)

Notice that the trend and scaling appear to be in reasonably good agreement

for the range of Reynolds numbers considered. For both results, the trend to

tend towards a limiting value seems to be confirmed.

11

3.3

Unsteady separation, Re=2000, α = 0.50

Unsteady separation requires a study on a very short amount of time. The

recirculation region grows with time from t=1.20 to t=1.90, figure 13. Then,

two different phenomena occur at the left and at the right of the bubble.

First, we observe a spike at the left of the main recirculation region at t=2.20,

figure 14. It occurs very quickly as we can see by comparaison with streamlines

at t=1.90, figure 13. Two new recirculations appear : first at t=2.20, under the

spike. This one will be the origin of internal vortex splitting.

Then at t=2.30, at the bottom of the spike itself. At t=2.40, figure 15, the

internal splitting occurs, increasing the spacing between the bottom recirculation and the main recirculation. This event leads to a clear combination of three

areas. Two vortices are about to split while a recirculation under them push for

it. Notice that the MRS model is coherent by looking at the zero-vorticity lines.

Indeed, at t=2.20, a zero-vorticity line errupts at x=0.9 where the streamlines

suddently bend. This erruption moves upstream as time evolves. From t=2.80,

figure 16, to t=3.00, figure 17, the position moves from x=1.0 to x=1.5. This

gap announce the beginning of internal vortex shedding. The location of a vortex splitting always occurs at a stagnation point (zero kinetic energy). It is also

along a line of zero streamwise pressure-gradient line. The process is explained

on figure.

Figure 14: Vortex splitting mechanism

12

Figure 15: t = 1.90, streamlines (left), vorticity countours (right)

Figure 16: t = 2.20, streamlines (left), vorticity countours(right)

Figure 17: t = 2.30, streamlines (left), vorticity countours(right)

Figure 18: t = 2.80, streamlines (left), vorticity countours(right)

Figure 19: t = 3.00, streamlines (left), vorticity countours(right)

13

3.4

Unsteady separation, Re=10000, α = 0.70

The same analysis is performed with a higher Reynolds number for the following

figures. Notice the reduced height of the separation bubble. Now, we focus on

the zero-vorticity field (red) erruption.

Figure 20: t = 2.40, streamlines (left), zero-vorticity field (right)

Figure 21: t = 2.60, streamlines (left), zero-vorticity field (right)

Figure 22: t = 2.90, streamlines (left), zero-vorticity field (right)

Figure 23: t = 3.10, streamlines (left), zero-vorticity field (right)

14

Unsteady separation for this reynolds number is a more complex phenomenon.

The upstream spike give birth to three inner recirculation regions at its bottom

as shown at t=2.60, figure 20. At t=2.60, figure 22, the situation is not straightforward. No internal splitting is about to occur and four different recirculation

regions are constricted in a small area. On the contrary, the adverse pressure

gradient also generates a downstream spike, see t=2.40, figure 23. Then, a splitting occur quickly at t=2.90, figure 25, and the first vortex shedds at t=3.10,

figure 26.

Figure 24: t = 2.40, streamlines (left), vorticity countours(right)

Figure 25: t = 2.60, streamlines (left), vorticity countours(right)

Figure 26: t = 2.90, streamlines (left), vorticity countours (right)

Figure 27: t = 3.10, streamlines (left), vorticity countours(right)

15

We sum up the temporal process with the schematic of figure 27.

Figure 28: Main events of high Reynolds number separation in the channel with

suction

3.5

Vortex shedding, Re=105 , α = 0.70

Vortex shedding is explained in [3] as a result of a convective instability leading to unsteadyness. The sinusoidal pressure gradient is amplified to allow an

internal vortex splitting and, then, the external vortex splitting. Finally, the

vortex shedds. Rather than trying to extract a mechanism based on pressure

gradient, we will try to analyse the process in terms of growing harmonics. We

will focus on the fourier transform of pressure.

The classical case is an alternate path of vortex on each wall, an example is

given on figure.

Figure 29: A vortex shedding case at Re = 105 , t = 15

16

The point we chose to collect pressure values over time is the first location

of extern vortex splitting. Let’s illustrate it for the first case (Re = 105 , α = 0.7

at t=5.6) :

Figure 30: Channel for vortex shedding study

We take the fast fourier transform of the pressure at this point from t=5 to

t=35. We obtain the plot figure 30.

Figure 31: Leading harmonics in the largest pressure collection time (t=5 to

t=35)

17

Note that there’s a leading frequency (f=1) but a lot of others cannot be

neglected. It means that the shedding is composed of different frequencies.

Let’s perform a frequency analysis at different times to see when this different

frequency show up.

3.6

Vortex shedding, Re=105 , α = 0.40

Figure 32: Frequencies represented t=5 to t=10 (left) and t=10 to t=15 (right)

Figure 33: Frequencies represented t=15 to t=20 (left) and t=20 to t=25 (right)

Figure 34: Frequencies represented t=25 to t=30 (left) and t=30 to t=35 (right)

18

3.7

Vortex shedding, Re=105 , α = 0.60

Figure 35: Frequencies represented t=5 to t=10 (left) and t=10 to t=15 (right)

Figure 36: Frequencies represented t=15 to t=20 (left) and t=20 to t=25 (right)

Figure 37: Frequencies represented t=25 to t=30 (left) and t=30 to t=35 (right)

19

3.8

Vortex shedding, Re=105 , α = 0.70

Figure 38: Frequencies represented t=5 to t=10 (left) and t=10 to t=15 (right)

Figure 39: Frequencies represented t=15 to t=20 (left) and t=20 to t=25 (right)

Figure 40: Frequencies represented t=25 to t=30 (left) and t=30 to t=35 (right)

20

3.9

Analysis of the results

For Re=105 , α = 0.40, we see two leading harmonics growing simultaneously

with time. The others can be neglected and the periodicity is thus composed of

two frequencies. See figure 41 for a schematic.

Figure 41: Harmonics evolution for Re=105 and α = 0.40

As suction ratio is increased, for α = 0.60, the number of main harmonics

increases and the analysis becomes more complicated. Three harmonics are

above the others, then one is selected to grow. Lastly, a bunch of harmonics are

growing quasi-homogeneously. See figure 42.

Figure 42: Harmonics evolution for Re=105 and α = 0.60

Finally, for α = 0.70, a high number of harmonics are represented. They all

grow up simultaneously. Finally, a slop accurs in the growth, see figure 43.

Figure 43: Harmonics evolution for Re=105 and α = 0.70

21

3.10

Strouhal number

We define the strouhal number as follows :

fL

(3)

V

We take α = 0.5, and we collect leading harmonics for each Reynolds number. This gives the frequency f in (3). V is always 1, L as well. It provides the

following plot (figure 44) :

St =

Figure 44: (Re,St) plot for α = 0.5

The trend seems to be universal. Indeed, in the cylinder case the curve is

the same qualitatively, see figure 45.

Figure 45: (Re,St) in the cylinder case

22

Now, we propose a study of the turbulence model used to predict separation.

4

Theory of turbulence modelling

The fundamental basis of fluid dynamics are the Navier-Stokes equations. The

incompressible form of these equations and the incompressible continuity equation are described as :

Dui

∂ui

∂ui

1 ∂p

∂ui

=

+ uj

=−

+ν

Dt

∂t

∂xj

ρ ∂xi

∂xj ∂xj

(4)

∂uj

=0

∂xj

(5)

and

The numerical study of this equation can be handled for reasonable Reynolds

numbers. However, for very high Reynolds numbers, a fine grid is required to

compute all the turbulent scales (DNS). Without parallel processing, this cannot

be done within reasonable computational time, that’s why we need to introduce

new tools to approximate the solution.

4.1

The Reynolds equation

In the RANS (Reynolds Averaged Navier-Stokes) approach to turbulence, all

of the unsteadiness in the flow is averaged out and regarded as part of the

turbulence. The flow variables, in this example one component of the velocity,

are represented as the sum of two terms:

ui (xi , t) = u¯i (xi ) + u0i (xi , t)

(6)

u0i

where u¯i (xi ) is the average of u at xi over time and

the fluctuation about

the time averaged value. We take the mean of both incompressibility and NavierStokes equation. For the first one, it’s straightforward :

∂ u¯j

=0

∂xj

(7)

Taking the mean of the incompressible momentum equation is not as straightforward because of the nonlinearity of the convective term. Taking the mean of

the left hand side of equation (1.1) gives us:

Dui

∂ui

∂ui uj

=

+

Dt

∂t

∂xj

(8)

Resulting from a classical simplification, we know that :

ui uj = u¯i u¯j + u0i u0j

Thus, the right hand side becomes :

23

(9)

∂u0i u0j

∂ui

∂uj

Dui

∂ui

=

+ uj

+ ui

+

Dt

∂t

∂xj

∂xj

∂xj

(10)

Since the mean velocity field is incompressible, it simplifies to :

∂u0i u0j

∂ui

Dui

∂uj

=

+ ui

+

Dt

∂t

∂xj

∂xj

(11)

The average of the momentum equation, called the Reynolds Equation, is

thus :

∂u0i u0j

∂ui

∂ui uj

1 ∂p

∂ui

+

=−

+ν

−

∂t

∂xj

ρ ∂xi

∂xj ∂xj

∂xj

(12)

We can write the Reynolds equation in its final version :

ρ(

∂ui uj

∂

∂uj

∂uj

∂ui

+

)=−

(−pδij + µ(

+

) − ρu0i u0j ))

∂t

∂xj

∂xj

∂xj

∂xj

(13)

On the left hand side we can identify the sum of three stresses : the first from

the mean pressure field, the second from the momentum transfer at molecular

level and, finally, the Reynolds stresses. The Reynolds stresses come from fluctuating velocity field. It is a second-order tensor composed of normal stresses

(diagonal terms) and shear stresses (off diagonal terms). The turbulent kinetic

energy k is defined as half of the trace of the Renolds stress tensor :

k=

4.2

1 0 0

ρu u

2 i j

(14)

The Turbulent-viscosity hypothesis

The Turbulent-viscosity hypothesis assumes that :

−u0i u0j = νT (

∂uj

∂uj

2

+

) − kδij

∂xj

∂xj

3

(15)

Where the positive scalar filed νt = νt (xi , t) is the turbulent viscosity. Using

this hypothesis in (9) leads to :

∂ui

∂ui uj

∂

∂ui

∂uj

1 ∂p

2

+

=−

[νef f (

+

)] − (

+ ρk)

∂t

∂xj

∂xj

∂xj

∂xi

ρ ∂xi

3

(16)

this model is very practical for computation. Unfortunately, for many flows

the accuracy of the model is poor. This shows that the physics of turbulence is

different from the physics of the molecular processes that lead to the relation for

the viscous stress in a Newtonian fluid. However, for simple shear flows, where

the mean velocity gradients and turbulence characteristics develop slowly, the

hypothesis is quite reasonable.

24

4.3

The energy cascade

Turbulence is composed of eddies of different sizes. The largest eddies of the flow

are unstable and break up, transferring their energy to smaller eddies. These

smaller eddies also break up and transfer energy to yet smaller eddies. This

energy cascade continues until the Reynolds number Re(l) = u(l)l

is sufficiently

ν

small so that the eddy motion is stable and molecular viscosity is effective in

dissipating the kinetic energy. Here l and u(l) are the characteristic length scale

and velocity scale of these stable eddies. This is important because it places the

dissipation of energy at the end of the energy cascading process. The rate of

dissipation, denoted , is determined by the first process in the sequence, which

is the transfer of energy from the largest eddies. These eddies are characterized

by the lengthscale l0 , the velocity scale u0 , the time scale τ0 = ul00 and have

energy of 12 ρU02 . Then the rate of transfer of energy can be supposed to scale

as uτ00 = ul00 . Consequently, scales as u3

l0 , independent of ν.

4.4

The Turbulent Kinetic Energy (TKE) equation

The TKE is obtained by multiplying the Navier-Stokes equations with ui , taking

the average, multiplying the Reynolds equation by ui , and substracting both.

Thanks to some averaging rules, we get :

∂k

∂ 1 0 0 0

∂k

1

∂k

∂ui

+ uj

=−

( u u u + u0 p0 − ν

) − u0i u0j

−

∂t

∂xj

∂xj 2 i i j ρ j

∂xj

∂xj

(17)

Where is the dissipation rate of turbulent energy, given by :

=ν

∂u0i ∂u0i

∂xj ∂xj

(18)

• The first term on the right-hand side is called turbulent transport and is

the rate at which turbulence energy is transported through the fluid by

turbulent fluctuations.

• The second term is pressure diffusion and is another form of turbulent

transport resulting from correlation of pressure and velocity fluctuations.

• The third term is the diffusion of turbulence energy caused by the fluid

natural mollecular transport

• The fourth term is known as production term. It represents the rate at

which kinetic energy is transferred from the mean flow to the turbulence.

4.5

The k − model

To be able to solve the TKE equation we need another equation. For this, we

introduce the gradient-diffusion hypothesis :

1 0 0 0

νT ∂k

1

u u u + u0 p0 =

2 i i j ρ j

σk ∂xj

25

(19)

Mathematically, it ensures that the solutions are smooth and that a boundary condition can be imposed on k everywhere on the boundary. Using this

model for the turbulent transport and the pressure diffusion, we end up with

the following model transport equation for k :

∂k

∂k

∂

νt ∂k

∂ui

∂uj ∂ui

+ uj

=−

[(ν +

)

] + νt (

+

)

−

∂t

∂xj

∂xj

σk ∂xj

∂xj

∂xi ∂xj

(20)

The same kind of law can be obtained for , empirically :

∂

∂ νt ∂

∂ui

∂uj

∂

2

+ uj

=−

(

) + C1 νt (

+

) − C2

∂t

∂xj

∂xj σ ∂xj

k

∂xj

∂xi

k

(21)

With the following values :

Cmu = 0.09, C1 = 1.44, C2 = 1.92, σk = 1.0, σ = 1.3

5

(22)

Influence of the constants on separation

The model clearly over-estimates the suction ratio required for separation to

occur. However it can be interesting to look at the effect of two main constants

of the model : κ and Cµ . This will be done for a high Reynolds number

(Re = 106 ) and suction ratio (α = 0.3).

5.1

The log-law

In the turbulent boundary layer, the log-law holds :

u+ =

1

log(y+) + B

κ

(23)

with :

uτ x2

δ

The value, commonly used but often discussed, is : κ=0.41

y+ =

26

(24)

5.2

Impact of a κ modification

We notice that modifying the κ value up to 36.7 % affects a lot the wall shear

stress but doesn’t change its minimum. That is to say that even a very consequent error on κ would not give a different separation prediction.

Figure 46: Effect ok κ variations on wall shear stress

5.3

The Cµ coefficient

The key coefficient in the definition of the turbulent viscosity is Cµ :

k2

(25)

For this value, the sensibility is extreme and even a 5 % decrease leads to

a big gap in the separation area as you can see on the wall shear stress plot,

figure 47.

The first computation gives a separation (because the wall shear stress goes

bellow zero) and the others don’t. This is a significant difference and shows

how important it is to have a high precision on this constant when studying

separation.

νt = Cµ

27

Figure 47: Effect ok Cµ variations on wall shear stress

6

Conclusion

In this research project, we worked on the characterization of incipient separation in term of adverse pressure gradient. The curve obtained matched the

asymptotical prediction of marginal separation theory. We showed different

events occuring at high Reynolds number : the evolution of upstream and

downstream spikes leading to the shedding of the first vortex. The study of

vortex shedding in terms of harmonics provided different behaviours, however

we noticed that the higher the suction ratio, the higher the number of leading

harmonics. Then, we connected the cylinder case with the channel with suction

case using the evolution of the Strouhal number with Reynolds number. The

same kind of curves are found. Finnally, we showed how the turbulence model

impacts the prediction of flow separation through the different constants.

28

References

[1] “Channel Flow with Suction” K.W.CASSEL

[2] “NavierStokes solutions of unsteady separation induced by a vortex”

A.V.OBABKO,K.W.CASSEL 2001 J. Fluid Mech.

[3] “Flow in partially constricted planar channels origin of vortex shedding and

global stability of Navier-Stokes solutions” M.E BOGHOSIAN 2011 Phd

thesis

29