Dean Flow, Separation, Branching, and Entrance Length

Louisiana Tech UniversityRuston, LA 71272

Slide 1

Dean Flow, Separation, Branching, and Entrance Length

Juan M. Lopez

BIEN 501

Wednesday, April 4, 2007


Slide 2

Dean Flow (Flow in a curved pipe)*This is an adaptation of Dr. Jones Spring 1997 Physiological Fluid Mechanics lecture on Dean Flow.

Although we call Dean Flow “Flow in a curved pipe”, technically it is flow in a torus, as shown below:

Figure 1 Dean Flow System Sketch, showing coordinate system

The center axis of the torus is a distance R from the center, C, of the torus. The coordinates within the torus are the radial distance (r) from the axis (A). The angle (ψ) between r and the vertical line through A, and the distance ξ along the axis.


Slide 3

Dean Flow – Governing Equations

• The equations for this are determined from the expansion of the steady Navier-Stokes equations:

• In this coordinate system, the Navier-Stokes Equations become

p1

uu


Slide 4

Dean Flow-Governing Equations

11

sin

cos11

sin

sin22

r

rrr

v

rr

v

r

v

rRrr

P

rR

v

r

vv

r

v

r

vv

momentumr

2

1

sin

sin1

sin

cos2

r

rr

v

rr

v

r

v

rRr

P

r

rR

v

r

vvv

r

v

r

vv

momentum

3

sin

cos11

sin

sin1

sinsin

cos

sin

sin

rR

vv

rrrR

v

r

v

rr

P

rR

R

rR

vv

rR

vvv

r

v

r

vv

momentum

rrr


Slide 5

Dean Flow-Governing Equations

• In conjunction with these, we must look at the continuity equation, , which becomes:

• Notice that if the radius of curvature is extremely large, so that 1/R -> 0, then the equations reduce to those for cylindrical coordinates.

0u

40

sin

cos1

sin

sin

rR

vv

rrR

v

r

v

r

v rrr


Slide 6

Dean Flow-Boundary Conditions

• The boundary conditions for this problem are:– Velocities must be finite for r->0– Velocity must be zero at the walls (no slip condition) i.e.

• Notice that these boundary conditions are separable. That is, they can be expressed in the form u evaluated along a surface = . This will impact the method used to convert the partial differential equation to a set of ordinary differential equations. For example, if the boundary condition on u were , we would have to seek solutions of a different form. On the other hand: would be a separable boundary condition and could be handled in a manner similar to that below.

arr vvv |0

rR

cos,, Ru

cos,, Ru


Slide 7

Conversion to a Linear Set of Partial Differential Equations

• As usual, the first question to ask is, “ can we get rid of the nonlinearities on the left hand side of the Navier-Stoked equation?”

• The technique Dean used was to assume that the solution will be a small perturbation on Poiseuille flow. – Certainly, if a/R is large enough, the torus

looks locally like a straight pipe, and the solution must converge to Poiseuille flow in the limit of a/R ->0.


Slide 8


• Dean, therefore assumed:

• He then assumed that are of order a/R, is of order 1, and that a/R << 1. From a practical standpoint, this means that any terms which are combinations of velocities, such

as , etc, will be of order and will be negligible. However, some of the convective accelerations will be maintained in the equation as a result of the Poiseuille flow term.

6/4/

522

pAP

vraAv

pvvv r

and,,,

22 raA

vvvv rr . 2/ Ra


Slide 9


• For example, in the ξ-momentum equation the first term (multiplied by a) is:

• The second (nonlinear) term is of order and will be neglected, but the first term is only of order a/R (because Aar is of the same order as and is thus maintained in the equation. However, the first term is linear and thus not as bothersome as the second.

72

22

r

vavrAav

varaAr

avr

vav

rr

rr

2/ Ra

22 raA


Slide 10


• Applying this analysis to the r-momentum equation, it will be clearer if the equation is first multiplied by a. With this, and with the substitution of equations 4 and 5, the left hand side of the equation becomes:

8

sin

sin2222

rR

avraA

r

avv

r

av

r

vav rr

r


Slide 11


• The first three terms are of order . For the third term, is of order a/R, and expands to three terms, which is of order 1, , which is of order a/R, and , which is of order . The only term of order a/R or greater is:

• Similar analyses can be performed on the right side of the r-momentum equation and on the other two momentum equations. The result is:

2/ Ra

sin

sin

rR

a

222vraA 2222 raA

vraA22

2v 2/ Ra

9

sin

sin222

rR

raaA


Slide 12


1011

sin

:2222

rv

rr

v

r

v

rr

p

R

raA

momentumr

1111

cos

:2222

rv

rr

v

r

v

r

p

rR

raA

momentum

1211sin

61

22

2

22

2

rr

v

rr

v

rr

v

R

rA

pArv

momentum

1301

:becomes continuity Similarly,

v

rr

v

r

v rr


Slide 13

Conversion to a Set of Linear Differential Equations

• The next step is to obtain a set of ordinary differential equations. To do this, we must break each of the functions into components which are individually functions of r, ψ, and ξ. For example, we would like to have , where the superscripts are used to distinguish between the parts of , which depend only on the indicated variable. That is, depends on r only, depends on ψ only, and depends on ξ only.

• First consider the ξ dependence of the dependent variables. Because the torus has circular symmetry about C, there is nothing to distinguish a cross section at one value of ξ from a cross-section at any other. Thus, the velocities cannot depend on ξ.

pvvv r and,,,

rr

rrr vvrvrv ,,

rrv

rrv

rv rv


Slide 14

Conversion to a Set of Linear Differential Equations

• It can be shown as follows that is a function of r and ψ only. Since the velocities do not depend on ξ, Equation 12 shows that is a function of r and ξ only. Thus, from simple integration, has the form , which can only happen if α is a function of ψ only. This is extremely important in terms of the solution to the partial differential equation. It means we can look for solutions to which are independent of ξ.

• Dean then uses an eigenfunction expansion technique (separation of variables) to convert the partial differential equations into a set of ordinary differential equations which can be solved. He assumes:

• Where, are functions of r only. These forms are then substituted into the Navier-Stokes equations to obtain ordinary differential equations.

• Since the ψ-dependence is explicity given in Equation 14, these ordinary differential equations will have derivatives of r only.

/p

p

p

,, rrp

pvvv r and,,,

14sin,cos,sin,sin '''' ppvvvvvv rr

'''' ,,, pandvvvr


Slide 15

Conversion to a Set of Linear Differential Equations 15

:''''2222

r

v

r

v

dr

dv

rdr

pd

R

raA

momentumr

r

16

:''''2222

r

v

r

v

dr

dv

dr

d

r

p

R

raA

momentum

r

1716

22

''

2

'2'

r

v

dr

dv

rdr

vd

R

ArArv

momentum

r

180

:'''

dr

dv

r

v

dr

dv

Continuity

rr


Slide 16

Solution of the Ordinary Differential Equations for vr and vψ

• The method by which these equations are solved is fairly interesting. First, the pressure can be eliminated from equations 15, and 16 to obtain the differential equation:

• Instinct would be to eliminate , but a better option is available.

1914 '''

2

22222

r

v

r

v

dr

dv

rdr

d

dr

dr

R

rarA r

''vorvr


Slide 17


• Think of this as an ordinary differential equation that describes the quantity defined by:

• The equation falls into the class known as equidimensional ordinary differential equations and has the solution:

,r

20,'''

r

v

r

v

dr

dvr r

216

3 2232'''

R

rarACr

r

B

r

v

r

v

dr

dvr


Slide 18


• Now, solve the continuity equation for , and substitute this into Equation

17. The result is:

• The solution for this is:

'' of in terms vvr

226

363

2232'

2

'2

RrarA

Crr

B

dr

dv

dr

vdr rr

23288

6

8

3

2

log 22422

2'

RrarACrrB

r

EDvr


Slide 19


• Substitution for this into the continuity equation gives :

• Now, we can apply our boundary conditions and solve for the individual velocity profiles.

'v

24288

730

8

3

2

log1 22422

2'

R

rarACrrB

r

EDv


Slide 20

Application of Boundary Conditions

• From our first boundary condition, B = E = 0. Otherwise, the velocity along the axis would be infinite. It remains to apply the non-slip boundary conditions. This is straightforward, and it leads to the following solution for :vandvr

25sin288

4 222222

R

raraAvr

26cos288

7234 4222222

R

rraaraAv


Slide 21

Solution for

• To obtain the differential equation for , combine Equation 25 with Equation 17. When this is done, the resulting differential equation is solved, and the result is as follows:

– An excellent example of Dean Flow solutions and an application that can be extremely useful from such flows is found in Gelfgat et al. 2003. The manuscript has been added to the course documents in case you are interested in looking at the work. The following figures stem from this work.

v

v

27sin10

341152

sin4

3

88662444226

2

3

2222

raraaraaraa

R

rA

raR

ArraAv


Slide 22

Streamlines

Figure 2 Streamlines for a variety of Dean Numbers [Gelfgat et al. 2003]


Slide 23

Streamlines

Figure 3 Description from manuscript of Dean Flow streamlines figure [Gelfgat et al. 2003]


Slide 24

Nondimensional form of the Equations, NonD Parameters

• It is instructive to nondimensionalized the equations for velocity. First, a good reference for velocity is the Poiseuille velocity at the center of the tube. In the notation used here (See Equation 5), this is:

• The Reynolds number (based on tube radius) is then:

• The nondimensional radius is:

2820 Aau

29Re32

0

AaaAaau

30'

a

rr


Slide 25


• The solutions for the three velocity components are then:

• Examination of Equation 34 shows that there are two nondimensional parameters which are important. One is a/R, which has already been talked about, and which determines the importance of the second term in square brackets.

31288

'4'1sinRe 2222

0 R

rara

u

vr

32288

'7'234'1cosRe 4222

0 R

rrra

u

v

34''9'211911520

sin'Re

4

sin'31'1 642

22

rrr

R

ar

R

arrv


Slide 26


• The other is , which determines the importance of the terms in curly brackets. The square root of this number,

• is called the Dean number. This can be used to determine, approximately, when the Dean’s flow solution is valid.

– By assumption (Equation 5), must be small, which means that the term in curly brackets in Equation 34 must be small. Thus, not only must a/R be small enough to make the terms it modifies be much less than 1, but also must be small enough to make the terms it modifies much less than 1. It will generally be more difficult to satisfy the condition on the Dean number than to satisfy the condition on a/R. For a relatively low Reynolds number, approximately 1000, the term that magnifies the curly brackets is about the same magnitude as a/R, so the terms in the curly brackets determine the validity of the solution unless the Reynolds number is very low.

Ra /Re2

35ReR

aDe

Ra /Re2


Slide 27

Relation to Textbook (5.4)

• The solution in your textbook is much more simplified explanation, and does not cover any of the derivation. The main items in the text that are of note are as follows:

• There is also a term, of much smaller magnitude, that applies due to the circulating flow within the torus. As opposed to Poiseuille flow where we have no ψ-circulation, we now have this circulation, and so must add a shear term:

3673728

cos1

4|

2

De

a

varrz

3772

sin1

4|

De

a

varr


Slide 28

Flow Separation (4.6)

• From Section 4.6 of your text:– Comes from Boundary Layer Theory– When flow reversal or recirculation occurs, the

boundary layer is said to be “Separated”.• The location at which the flow first reverses is known as the

“Separation Point”• The location where the fluid again moves in the same

direction is known as the “Reattachment Point”.

– In biological systems, flow separation arises in lungs and blood flow.

• Implicated in artherosclerosis


Slide 29

Flow Separation (4.6)

• When a change in cross sectional area occurs, the velocity must change.

• To compensate for the momentum balance, the pressure must change.

• A positive pressure field rather than a decreasing pressure due to flow is a necessary, but not sufficient, condition for flow separation.

• Flow reversal occurs when the adverse pressure field is large enough to overcome the viscous forces at the wall and the inertial components of the fluid elements.


Slide 30

Branching Flows (5.5)

• Branching flows exist everywhere in blood flow analysis.

• Flow separation is a major part of the study of branching flows.

• As was mentioned before, flow separation is implicated in the formation of atherosclerosis. This is illustrated in several of the branching flow diagrams in your textbook.


Slide 31

Entrance Length (5.3)

• We often talk quite a bit about “fully developed flow”.

• This occurs after the boundary layers have converged or become asymptotic.

• However, we can estimate the entrance length required to achieve fully developed flow in a simple circular pipe by:

38Re113.0 RXC


Slide 32

Entrance Length (5.3)

• Equation 38 is for steady flow only.

• For unsteady flows, including pulsatile flows, this length can vary and exhibit oscillations.

• Flow in the major arteries is not fully developed.

• Thus, these flows are inherently two or three-dimensional and sensitive to the inlet flow conditions.


Slide 33

References

• Gelfgat, A.Y., Yarin, A.L., Bar-Yoseph, P.Z., (2003), Dean vortices-induced enhancement of mass transfer trough an interface separating two immiscible liquids, Physics of Fluids, Volume 15, No. 2, pp. 330-341


Slide 34

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Dean Flow, Separation, Branching, and Entrance Length

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