Peube J-L. Fundamentals of fluid mechanics and transport phenomena

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General Properties of Flows 323

Finally, we must fix the initial upstream condition for the velocity distribution at

a given abscissa often taken as the origin

= 0. As we have explained in section

6.5.3.3, the stagnation point A of the flow of a inviscid fluid over the obstacle

(Figure 6.17) is the departure point of the boundary layer

that

can be easily

calculated in this zone where its thickness is quite small ([SCH 99], [YIH 77]).

the leading edge of the obstacle is of negligible thickness

(plane or wedge-

shaped wall), the thickness of the boundary layer is here taken to be zero.

6.5.3.6.

General properties of boundary layers

6.5.3.6.1.

Physical interpretation of the boundary layer

We can immediately note that

the condition of zero velocity on the wall also

leads to a condition of zero acceleration

. Conversely,

the viscous stress becomes

zero at the outer edge of the boundary layer

. This is therefore a zone where the

pressure gradient, which is constant, sees its action balanced by the acceleration in

the outer region and by the viscous friction at the wall. This situation is shown

schematically in Table 6.1 which indicates the dominant terms following the height

in the boundary layer.

The solution previously obtained in section 5.4.5.4 for the heat equation is of the

same kind as that of the boundary layer, with time replaced by the

coordinate. The

zero-velocity condition imposed on the wall leads to

viscous diffusion of the viscous

stress and of the vorticity.

This results in an augmentation of the boundary layer

thickness as an abscissa function, as indicated by the order of magnitude



ULL

UPG

found earlier [6.92].

height in the

boundary layer

acceleration

quantity

pressure

gradient



viscous stress

outside (inviscid

fluid)

X X 0

inside boundary

layer

x X x

wall 0 X X

Table 6.1.

Balance of “forces” in the boundary layer

324 Fundamentals of Fluid Mechanics and Transport Phenomena

This diffusion can be counterbalanced by an aspiration of fluid across the wall.

Consider the simple case of a plane wall placed in a uniform flow of velocity

and

realize a suction of fluid across the wall at constant velocity



(Figure 6.18). We

see immediately that the solution:



exp [6.96]

satisfies equations [6.91] of the boundary layer and corresponds to an established

solution where the thickness

is constant. Figure 6.18 represents this flow.

Figure 6.18.

Established boundary layer with suction

through a wall in an uniform flow

We note that in the case where we have blowing rather than suction (



positive), the solution is not acceptable: the boundary layer no longer exists if the

blowing is sufficiently strong ([SCH 99], [YIH 77]).

6.5.3.6.2.

Non-dimensional equations

Let us write equations [6.91] in non-dimensional form with the following change

of variables:



ULxuuLyReyLxxUvRevUuu

2121

This gives:

;







[6.97]

Equations [6.97] do not contain the parameter

Re . This results in the non-

dimensional solution

being independent of the Reynolds number

. So, all bodies which

have the same form have identical velocity distributions



at the wall; they therefore

General Properties of Flows 325

have the same boundary layer development. We note however that the Reynolds number

comes into play as soon as we recast the problem using dimensional data for the

coordinate

and notably for the thickness of the boundary layer (formula [6.92]).

6.5.3.7.

Separation of steady flows

The preceding discussion shows that, in the external part of the boundary layer,

the flow is close to the flow of an inviscid fluid where the pressure gradient is

determined by Bernoulli’s theorem: the modulus of the velocity decreases if the

pressure increases.

Close to the wall, the weak kinetic energy no longer plays an appreciable role

and the variations of the viscous stresses are opposed to the pressure gradient in the

boundary layer equations. The flow which is nearly purely viscous close to the wall

results from two causes:

1) The

pressure gradient

leads to a tendency of the flow towards decaying

pressures as in a Poiseuille flow (section 3.4.2.5). The velocity curvature profile at

the wall is equal, according to [6.91] and [6.95] to:



2) The

external velocity



imposed creates a viscous entrainment in the

boundary layer in a manner analogous to the corresponding phenomena in a Couette

flow (section 3.4.2.3).

When the flow external to the boundary layer



increases with the abscissa

, the longitudinal pressure gradient is negative and the velocity curvature profile

has a constant sign (Figure 6.19a). In the opposite case (Figure 6.19b), the sign of

the curvature of the profile changes, and this can lead, close to the wall, to a flow

which is reversed with respect to the external flow



(a) dp/dx < 0

(b) dp/dx > 0

(x) increasing

(x) decreasing

Figure 6.19.

Velocity profile near the wall with a

(a) negative or

(b) positive pressure gradient

326 Fundamentals of Fluid Mechanics and Transport Phenomena

The appearance of a flow from downstream to upstream is in

contradiction with

the parabolic properties

of the boundary layer equations: the velocity distribution in

the zone of reversed flow no longer depends only on the upstream conditions the

external velocity

(x) considered, but also on the downstream conditions. The

boundary layer equations are thus no longer applicable in such a zone.

(a) (b)

wake

rotational

Figure 6.20.

(a)

Separation with reattachment in the boundary layer (profiled obstacle);

(b) separation and creation of an open wake (unprofiled obstacle)

There therefore exists a back stagnation point S in a viscous flow where the

friction stresses on the wall are zero (Figure 6.20a). In fact, the flow, which is

reversed with respect to the main flow, comes from a rotational zone which can be

manifested in two forms:

– either a relatively thin rotational zone is generated within the boundary layer,

but reattaches at point R (Figure 6.20a); following what was said above, this can

only happen if the effect of the pressure gradient is sufficiently weak for the viscous

entrainment to constitute the dominant effect. This is the case for

profiled obstacles

whose reducing section in the downstream direction is very gradual;

– or a completely open wake can be generated downstream of an obstacle which

is not profiled (Figure 6.20b).

Note that the

position of the separation point is independent of the Reynolds

number

, provided the velocity distribution of the inviscid fluid at the wall does not

depend on the Reynolds number either. This is the case for the irrotational inviscid

fluid around a profiled obstacle which is not modified (to second order) and

obviously does not depend on the viscosity. For the unprofiled obstacle, the real

velocity field is different from the field calculated in an inviscid fluid on account of

the presence of a rotational wake which is fed from downstream; however, if we

assume that the structure of this wake is independent of the Reynolds number, we

see that it is consistent to assume that the position of the separation point is also

General Properties of Flows 327

independent of this parameter: this is confirmed by experiments,

provided the

boundary layer is not turbulent

([SCH 99], [YIH 77]).

6.6. Unsteady flows and steady flows

6.6.1. Introduction

The temporal evolution of the properties of matter is fundamentally based on the

balance laws of the associated extensive quantities. We have already discussed in

Chapter 2 the difficulties of representing the continuous medium which we

encounter depending on whether we choose to use a Lagrangian (substantial)

description of the fluid particles or a Eulerian (spatial) representation of the flow.

We must now return to the fundamental difficulties which arise when we use

Eulerian variables.

The fields to which matter is subjected are furthermore always due to actions at a

distance performed by other material elements: a gravitational field is caused by the

presence of mass, an electric field results from the presence of charges, an

electromagnetic field is due to electric charges in movement at either the

macroscopic or the microscopic scale. A field is described by functions of space-

time variables in a reference frame (known as the laboratory reference frame)

associated with a flow device or an object moving with respect to a fluid (vehicle,

plane, etc.). There are numerous situations for this observer in which the velocity

fields and the material quantities are not functions of time, but only of space. The

corresponding phenomena are therefore steady. This

terminology only has meaning

in reference to this privileged reference frame, the quantities attached to the

material particles being always functions of time

(Lagrangian representation).

However, these steady phenomena, when they exist, always arise as a result of

the evolution of a transitional regime. Thus, in many situations, the transitional

regimes do not lead to steady flows and we observe complex phenomena which we

will describe very briefly here.

In order to simplify the discussion, we will consider in what follows

an inviscid

or Newtonian fluid of constant density

, unless otherwise stated. The variations of the

physical properties, if they are not too great, do not significantly modify the

structure of the phenomena which we will discuss.

We will leave aside questions related to the existence and to the uniqueness of

solutions of the Navier-Stokes equations, the understanding of which requires a

more advanced course in mathematical analysis. In this domain many questions

328 Fundamentals of Fluid Mechanics and Transport Phenomena

remain open, and the physical aspects of the phenomena which we describe in this

textbook demonstrate the formidable complexity of such an eventual theory.

6.6.2. The existence of steady flows

For a flow and the associated transfers to be steady, it is necessary for the

boundary conditions describing the corresponding problem to be steady; in

particular,

actions on the flow by fixed elements in this reference frame should be

independent of time. For example, the flow between upstream and downstream

infinite reservoirs at constant pressure and connected by a nozzle (section 5.5.4) can

be independent of time if the system is described in the reference frame of the

nozzle. However, before being observed at a given flow rate, the flow was created

from a zero pressure difference and it followed an evolution through the following

states: a subsonic regime with increasing velocity, then a sonic regime and finally a

supersonic regime with the progressive appearance of a shock wave which descends

in the divergent part of the pipe until the pressure gradient is stabilized.

The solution of the problem defined by the steady boundary conditions is not

always unique. The nature of the boundary conditions to be used is often a source of

considerable difficulty (section 5.6.3). Consider for example the flow between two

cross-sections

and S

> S

) in a divergent pipe (Figure 6.21). Under the

assumption of an inviscid incompressible fluid and with the approximation of the

flow by slices, Bernoulli’s theorem can be written:

 

The flow rate

VSq

in this section of the pipe can be immediately

obtained:

2 1



with: .



There exist two opposite flow rate values for this steady flow. We will see later

that only one of these is really acceptable in the context of the preceding

assumptions.

Let us now perform an experiment with a plane pipe whose divergent has a

sufficiently large angle (of the order to 10 degrees or so). Here we observe that the

steady flow follows one of the walls while it separates from the other. The flow

chooses the wall to follow as a result of particular circumstances of the transitional

General Properties of Flows 329

regime. For example, in the configuration in Figure 6.21a, we have included in the

walls two suction orifices A

and A

which are not activated in the steady regime

and which we can use in order to create a small additional transitional depression in

order to “suck” the flow towards a chosen wall. This suction does not need to be

constant: once the flow has attached to one of the walls, it will remain so after the

suction has been stopped. Such flows with two stable positions can be used to

control flows in fluid circuits (fluidics command and control).

suction orifice A

(a) (b)

Figure 6.21.

Steady flows with two configurations

For the two preceding flows, the pressure is nearly constant in each of the

sections S

and S

. Writing the boundary conditions for the steady flow of a viscous

fluid leads to considerable difficulties, because there no longer exists a velocity

potential and we know that we cannot independently specify the entry and exit

conditions of a flow since the transport properties on the trajectories are dominant

(section 5.6).

Figure 6.21b shows another configuration where two flow structures in a pipe are

possible depending on whether the flow reattaches or not on an obstacle placed on

its wall. The existence of reattachment is also related to the transitional regime

which leads to the fully established flow. Examples of the same kind exist for sheets

of water over spillways, which may flow above cavities either ventilated (i.e. filled

with air) or not ([CHA 04] p. 399, [JAI 01] p. 264).

The preceding problems are examples of systems presenting hysteresis (the state

of a system depends on its history). From a physical point of view, we can note that

we have here a “retroaction” in the upstream direction which leads to the existence

of a memory for the flow.

330 Fundamentals of Fluid Mechanics and Transport Phenomena

In summary, a flow problem with steady boundary conditions does not

necessarily have a unique steady solution. We could also say that a steady problem

is not necessarily a well-posed problem.

6.6.3. Transitional regime and permanent solution

6.6.3.1.

Relation between pressure and flow rate in a fixed stream tube

Depending on the imposed conditions, pressure can be the cause or the

consequence of movement of a fluid. Aside from cases where the viscosity plays a

dominant role (low Reynolds number flows), the pressure (or the driving pressure

depending on the case) balances the acceleration, and for steady flow of an inviscid

fluid, Bernoulli’s first theorem treats the pressure (or the total enthalpy) as a

component of the total mechanical energy which is conserved for a fluid particle.

This local property no longer exists in unsteady flows.

Consider the flow of an inviscid fluid of constant density whose fluid trajectories

are fixed; they are thus coincident with the streamlines (and the emission lines). A

stream tube is therefore a surface on which the inflow and outflow of fluid occurs

over the cross-sections S

and S

with velocities V

and V

(Figure 6.22). The

assumption of incompressibility leads to a volume flow



SVtq

circulating in

the stream tube which is independent of the cross-section used to evaluate it; we will

treat it thus as a variable. Let us apply kinetic energy theorem [6.85] to the interior

domain D of the stream-tube element limited by the surfaces S

, S

and S

(Figure

6.22); we have:

³³³





DSS

dsnV

pdsnV

pdv

222

UUU

where p designates the driving pressure by way of simplification.

(a)

(b)



Figure 6.22.

Flow by slices inside a stream tube:

(a) convergent tube; (b) divergent tube

General Properties of Flows 331

In the slice approximation of the flow, the quantities are uniform in all cross-

sections, and after simplification by q

, we immediately obtain:



Let:

and



The relation between the flow rate q

and the pressure difference p

 p

can be

written:

ppqB

  [6.98]

This equation shows that the pressure is balanced by two acceleration terms, one

unsteady,

, and the other,

qB , corresponding to the steady flow,

proportional to the square of the flow rate and independent of the direction of the

flow. This separation is related to the Eulerian description of the phenomena, and

from a physical point of view there are not two kinds of acceleration for the fluid

particles. As we have already said (section 3.3.3.5), this description is always

associated with a favored reference frame resulting from the existence of boundary

conditions on the particular surfaces which are here the sides of the stream tube.

We should note that the coefficient A of equation [6.98] has an order of

magnitude proportional to the length, contrary to the coefficient B which only

depends on the values of the inflow and outflow sections. Furthermore, the 1/S

dependence of the integral A shows that a severe intermediate narrowing will not

modify the values S

and S

but will lead to a considerable increase of the coefficient

A, in other words to the pressure difference necessary for the transitional

acceleration in the Eulerian representation. The reason for this is the existence of a

strong acceleration in sections of small dimension where the velocity takes on high

values in order to conserve the flow rate.

6.6.3.2.

Properties of the solutions

Let us take B to be positive (

SS ! ), which does not restrict the generality of

the reasoning. Equation [6.98] can only therefore possess steady solutions if the

difference p



is positive. In this case, we have the two steady solutions

332 Fundamentals of Fluid Mechanics and Transport Phenomena



Bppq

r , corresponding to an established flow, either in the positive

direction, or in the negative direction of the axis

(section 6.6.2). However, the

possibility of realizing such a solution depends on the existence of a transitional

regime which can lead to this kind of situation.

In the unsteady regime, the pressure difference p



can be negative. However,

an examination of equation [6.98] shows that if it is always negative, the same goes

for the derivative

; we can thus see that the flow rate q

decays indefinitely,

which is physically unacceptable.

Suppose now that the difference p

 p

is positive and let

  

BpptQ



; Q(t) is the positive value of the flow rate under the

assumption that the unsteady term is negligible. Equation [6.98] can be written:



qtQB

A  [6.99]

We see that:

– if the instantaneous flow rate q

is greater than Q(t), the derivative

negative: the flow rate q

decays and approaches the value Q(t);

– if q

lies between – Q and + Q, it therefore increases, and approaches the value

Q(t) once again;

– if the instantaneous flow rate q

is less than  Q(t), the derivative dtdq

negative: the flow rate q

decays and moves away from the value  Q(t).

This property leads to the flow rate being bounded if the positive quantity Q is

bounded. If Q(t) tends to a limit for infinite t, the same goes for the instantaneous

flow rate q

if the latter remains always greater than – Q. Under the assumption of

constant pressure difference p

 p

, we see that the transitional regime does not

allow the solution

  

BpptQq

v 21

  to be attained.

The inviscid fluid model does not apply for negative flow rates, and the viscosity

must be introduced. The study of viscous fluid flow between two infinite divergent

planes can be performed (the Jeffery-Hamel exact solution of the Navier-Stokes

equations). We find that a separation occurs for a Reynolds number which tends to