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

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Transport and Propagation 233



This equation is encountered in problems of potential flow of incompressible

fluids, and in problems of thermal conduction or mass diffusion in steady flows. The

presence of a right-hand side (Poisson equation) implies the existence of sources for

the quantities studied.

Constant coefficients hyperbolic equations with no right-hand side take the form

of the wave equation:



in which the y variable is often the time. The characteristic curves are the straight

lines constant  xy

[

and constant  xy

By performing the change of

variables



[

,, oyx , the equation can be written:

Its general solution can be written using two arbitrary functions ) and <:

       

yxyxf <) <)

[

The reduced form of the parabolic equation is the heat equation:



in which the y variable is very often the time.

5.4.5. Second order partial differential equations and their boundary conditions

5.4.5.1. Introduction

As physical problems do not usually occur in the form of a Cauchy problem, on

account of the presence of boundaries (walls or other surfaces) surrounding more or

less completely the fluid domain under study, and on which we must impose specific

234 Fundamentals of Fluid Mechanics and Transport Phenomena

conditions. The properties discussed in the preceding sections regarding the

characteristic curves or surfaces show that these depend on the nature of the system

of partial differential equations. We will first examine the case of a quasi-linear

second order equation with constant coefficients.

5.4.5.2.

Elliptic equations

The Cauchy problem, which involves specifying the function values and its first

derivatives on a curve C

, generally allows the unique determination of the solution

in the neighborhood of this curve C

. However, physical problems are not given in

this way for elliptic equations, and we generally have a problem posed in a domain

on the boundary of which are given either:

– the unknown function (Dirichlet problem);

– the function’s normal derivative at the boundary (Neumann problem);

– or a mixed condition in the form of a linear relation between the function and

its normal derivative (mixed problem).

Problems involving heat conduction in a steady flow regime have already been

discussed in section 2.3.2. Steady subsonic flow of an inviscid fluid about an

obstacle is a Neumann problem (Chapter 6).

Elliptic problems cannot deal with propagative phenomena; they can generally

only represent steady phenomena or unsteady situations where propagation plays a

negligible role (domains which are small compared with a wavelength for example).

We will come back to discuss this point on numerous occasions.

In the case of the Neumann problem, a compatibility condition exists as a result

of Otrogradski’s theorem. For example, the Poisson equation of a steady 2D heat

conduction problem in a domain

D of the plane (x,y):





Tgraddiv

imposes on the curve C, which bounds D, the integral condition:



³³³³

DCC

dsdsTgraddivdnTgradd

VOOO

A .

This expresses the global conservation of energy, where the thermal flux

crossing the curve C must be equal to the thermal power generated in D under

steady conditions.

Transport and Propagation 235

Under usual conditions, the solution of an appropriately posed problem, of one

of the preceding kinds (Dirichlet, Neumann or mixed), is unique and elliptic

equations do not lead to difficulties, provided of course that they correctly represent

the physical problem studied.

5.4.5.3.

Hyperbolic equations

For equations of this kind, the discussion of section 5.4.2 outlined the ideas of

propagation and influence domain. The concept of the Cauchy problem is here

degenerated to a certain extent, due to the structure of the solutions. As already

discussed in section 5.2.4, the initial data have no influence in transverse directions

with respect to the characteristic curves.

In addition to the initial conditions, we often impose other boundary conditions.

For example, the presence of an obstacle in a flow leads to the conditions

0 ww nf on its wall, associated with the equation for the velocity potential f. This

condition leads in particular to a reflection of waves on the considered wall, an idea

which we will encounter for diverse phenomena (acoustic, supersonic flows, etc.).

In practice, the Cauchy problem is rather academic, as physical problems are

always posed in limited domains at the boundaries of which the boundary conditions

are generally defined by external data. However, its study (section 5.4.2) has

allowed us to define the characteristic curves and to show that these strictly limit the

influence domain of conditions given on a bounding curve. Conversely, the value of

an unknown function at a point M cannot depend on data outside the influence

domain formed by the characteristic curves which arrive at this point (Figure 5.5).

The notion of information transmission is closely associated with these

considerations: the characteristic curves constitute the means of information transfer

in the medium considered.

A B

(a) (b)

Figure 5.5.

Domain D influencing the value of the unknown function f in M

The boundary conditions associated with the wave equation can often be

considered as initial conditions. However, the direction of travel of the characteristic

236 Fundamentals of Fluid Mechanics and Transport Phenomena

curves is not important for a wave equation which is invariant if the time direction is

changed (section 1.1.1.3). Figure 5.5a shows an example of initial data for an

acoustic problem (given at time t = 0), while Figure 5.5b corresponds to velocity

data given for a supersonic steady flow in the plane (x,y) (which can be reduced

under certain particular conditions to a wave equation).

5.4.5.4.

Parabolic equations

We have seen in section 5.4.2 that a double characteristic curve exists on which

there is no longer a propagation relation. Taking it as a coordinate curve, we obtain a

heat equation [5.50] (see section 5.4.3) whose behavior we will study (we let t be the

parabolic variable y which we will interpret as time):

[5.50]

We have already seen, in section 1.1.1.4, the consequences of asymmetry

between the variables x and y; in particular, the variable y must evolve towards

increasing values (from past to future, or from upstream to downstream, etc.).

We will now treat the problem of a thermal shock

in a semi-infinite medium. We

will let f be the temperature in equation [5.50] and consider the thermal problem

which corresponds to a unit temperature step function imposed at the initial instant

t 0, at the origin x 0 of a semi-infinite medium )0( tx , initially at temperature

f = 0. We impose therefore the boundary conditions:



1),0(00

00,00

tftx

xftx

[5.51]

The scales X and Y corresponding to the variation F of f in [5.50] are:

The preceding estimate leads to the definition of the variable

tx 2

and

we look for a solution of the form

 

txf , . Substituting this function into

equation [5.50], we have:

5 The word “shock” is relative here to an initial condition and not to a discontinuity appearing

in the solution.

6 We have here a self-similar solution, i.e. which is invariant with respect to a group of affine

transformations. Flow equations, which are exact or approximate for high Reynolds numbers,

can have such solutions ([SCH 99], [YIH 77]).

Transport and Propagation 237

0'2''  ff

from which we obtain:







exp'

 Af

Taking account of boundary conditions [5.51] leads to:

( , ) exp (- ) 1 ( 2 )

fxt du erfx t





[5.52]

with:

exp (- ) ( 1)

erf x du erf



d

Figure 5.6 shows the temperature diffusion imposed at the origin, whose

influence can be felt in a zone, whose width, in the order of X, is proportional to t

1/2

(curves 1 to 4 for different increasing times). The dimensional presentation of the

problem will be treated in section 8.3.2.2.2.

2 3

Figure 5.6.

Influence zone of a unit step function (of temperature) imposed at the origin

We thus see that the behavior of the parabolic differential equation lies

somewhere between that of the hyperbolic and elliptic equations:

– the double characteristic curve is a preferred axis for the transmission of

information;

– the form of solution [5.52] shows an instantaneous action at all points of the

axis Ox: the notion of propagation has completely disappeared;

– we have “diffusion” (transverse spreading) of the initial data about the

characteristic curve.

238 Fundamentals of Fluid Mechanics and Transport Phenomena

The same initial boundary conditions applied to the wave equation would lead to

an acceptable solution for negative values of y, which is not the case here: the heat

equation describes an irreversible phenomenon, in other words the impossibility of

reversing the time direction (section 1.1.1.3).

5.4.5.5.

Mixed equations

In section 5.4.2 we saw that, depending on the quantity

BAC  , quasi-linear

partial differential equation [5.43]:



can be either elliptic, hyperbolic or parabolic. The nature of the solution depends

therefore on the local coefficients value. The simplest example of a mixed equation

is the Tricomi equation:



For negative values of x this equation is elliptic, while for positive values, it is

hyperbolic. It is a summary model of the situation described in section 5.3.2.3,

where, in a subsonic flow, the problem is elliptic, while a supersonic flow has a

hyperbolic character. The passage from a subsonic flow to a supersonic flow

happens in a transonic range, which may be locally represented by a Tricomi

equation (the passage from the reduced form of the elliptic equation to the reduced

form of the hyperbolic equation assumes that a coefficient changes sign when going

to zero):

elliptic

zone

hyperbolic

zone

Figure 5.7

Mixed problem (Tricomi equation)

Consider a flow such that it is subsonic for negative values of x and supersonic

for positive values. We assumed that the speed of sound is attained by the flow

Transport and Propagation 239

velocity at x = 0. Determination of the solution in the elliptic zone is possible using

the boundary conditions on the curve C

(Figure 5.7) and the condition

 BAC here corresponds to sonic velocity on the boundary x = 0 between the

subsonic zone and the hyperbolic zone. In the supersonic zone, the flow is

determined by the upstream conditions. So, as any real subsonic flow is generated

by an ensemble of conditions upstream and downstream, the presence of a

supersonic zone leads to a contradiction corresponding to the presence of shock

waves, which we will now examine.

Note that the coefficients of partial differential equation [5.42] are not only a

function of the coordinates, but also of the unknown function and its first

derivatives. Contrary to the Tricomi equation, the nature of the equation and the

(associated) boundary conditions that must be associated are not easily predictable

in general, as they depend on the solution values.

5.5. Discontinuities: shock waves

5.5.1. General considerations

The presence of shock waves may feature in the varied conditions of supersonic

flows of compressible fluids. We will show, using examples, that the existence of

characteristic curves or surfaces and the nature of the solutions to the hyperbolic

equations lead to the possibility of discontinuities.

We will first of all study two examples (1D flow of an inviscid compressible

fluid and steady supersonic flow) for which a discussion of the characteristic curves

leads to the impossibility of a continuous flow. We will then consider a steady 1D

flow of an inviscid fluid in a nozzle which is governed by a differential equation

which again shows the necessity of shocks.

5.5.2. Unsteady 1D flow of an inviscid compressible fluid

Let us take the example of a one-dimensional flow of a compressible fluid

verifying equations [5.13] (section 5.3.2.1). The characteristic transport curve

utx

ensures constant entropy at all points, which leaves only two

independent variables p and u. This condition leads to the relations:

 

JJJJ

JUJ

;const

),(



txp

240 Fundamentals of Fluid Mechanics and Transport Phenomena

in which (p

) are the initial conditions of the fluid for a zero velocity.

Let us now apply the method of characteristics on a grid discretized as defined in

section 5.4.2. Starting from two neighboring points R and S whose physical

quantities (p

, u

) and (p

, u

) are known, and which are not located on the same

characteristic curve (Figure 5.8a), we trace out two characteristic curve segments

with respective slopes u c r which intersect in plane (x, t) at the point T whose

the values of the unknowns (p

, u

) are calculated using the relations

0 r ucp

(with notations of section 5.4.2) taken on the previous curves

segments. These are written:

 

0;0  

STSSSTRTRRRT

uucppuucpp

These relations can be used for calculation of the values (p

, u

) for pressure and

velocity at point T. At first order, we can take ccc

SSRR

 , so that we have:

 

111

TRS SRTRS SR

pp cu u u u u p p

 c

¯

¡°

   

¢±

¡°

¢±

We can then deduce from thermodynamic relations with constant entropy, the

specific mass

and the sound velocity c

at point T.

Let us consider the case where pressure and velocity take uniform values on the

characteristic curve issued from point R, which having then a constant slope c + u, is

a straight line. If R' is a neighboring point of R on this characteristic curve, at point

T and at the intersection T' of characteristic curves issued from R' and S, we have:







.0;0

''''

 

RTRTRTRT

uucppuucpp

Taking out the first relation from the second relation, we obtain the relation on

characteristic curve of slope c + u: between points T and T':







TTTT

uucpp

So, between points T and T' we obtain the relation on characteristic curve of

slope c + u:







TTTT

uucpp

We then obtain the relation:

Transport and Propagation 241





 

TTTT

uucpp

which shows that pressure, velocity and sound velocity values are the same at points

T and T'. Going on from place to place on characteristic curve of slope c

+ u

issued from S, we see that the flow properties are identical along this curve, which is

also a straight line. The corresponding flow (a simple progressive wave) is then

characterized by the propagation along axis Ox with velocity c

+ u

of pressure 'p

and velocity 'u variations between the two characteristic curves going through R

and S. These variations verify the relation:

ucp ' '

The previous calculation can be continued on the next characteristic curves of

the same family which will possess the same property of simple progressive waves.

The reader can easily show that another family of simple progressive waves exists

associated with the other family of characteristic curves of slope  c + u. For these

waves, propagating towards the negative part of axis Ox, we have the relation

ucp ' '

(a)

x = c

t + OA

(b)

Characteristic curves

Figure 5.8.

(a) Application of the characteristics method and evolution of the

characteristic straight lines for a decreasing pressure (or a negative velocity);

(b) evolution of the velocity at the points O, C and F

Now consider the domain formed by the first quadrant of the plane (x, t) with the

following boundary conditions:

 

     

tUtutPtpxt

xupxpxt

,0,00:0

00,0,0:0

242 Fundamentals of Fluid Mechanics and Transport Phenomena

The domain thus defined is the influence domain of the given conditions on the

positive sides of the axes Ox and Ot. The positive side of the axis Ox is at rest at the

initial instant and we apply pressure and velocity variations at the origin x = 0, as a

function of time, which will then propagate on the Ox axis.

As previously, we can apply the method of characteristics by starting from the

points O, A, B, C, D, E,…,P, Q, etc., of the Ox and Ot axis (Figure 5.8a), and

calculating the values of the unknowns p and u using the relations 0 r ucp

taken on appropriate segments of the characteristic curves of slopes .

c+ u

However, the initial state, being uniform and at rest, can be considered as a

regime of simple progressive waves. Indeed, by applying previous relations on

characteristic curves issued from points A, B, C, D, E, etc., we see that the initial

rest data are transmitted to points H, I, etc., on the straight lines issued from

previous points on all parts of the plane above the straight line x = ct + OA.

Previous results show that propagation can only proceed with simple waves for

increasing times. It can be verified that the values (p

, u

) at point O are also

obtained from place to place for all points of the characteristic curve x =c

t issued

from O, on which the fluid state is uniform. Calculation is then carried out at points

P, Q, etc., of axis Ot, and it can be seen that we have a flow with simple progressive

waves, propagating on the characteristic straight lines of a positive slope.

The difference of pressure and velocity values between two neighboring

characteristic curves verify the relation

ucp ' '

of simple progressive waves.

This condition must be verified at points P, Q, etc., of axis Ot, so as not to create an

inverse propagation towards negative times on characteristic curves of negative

slopes: this should be not acceptable according to the physical aspects of problem.

This compatibility condition is a consequence of the definition of the initial data

curve itself, since there exist characteristic lines of a negative slope issued from axis

Ox intersecting axis Ot, which involves the relation ucp ' '

The simplest practical realization of the preceding flow involves imposing a

suitable velocity on the fluid matter by means of a moving piston. We will leave it to

the reader to determine the domain thus defined in the plane (x,t) for this Lagrangian

condition (we consider the trajectory of the piston).

When the pressure and velocity variations are negative and decreasing

(expansion), the slope of the characteristic lines is a decreasing function of time

(Figure 5.8a). These divergent straight lines thus form the shape of a fan, which

involves a broadening of the wave front as it propagates; Figure 5.8b shows the form

of the temporal velocity variation at the origin O, and then at points C and F.