Matematick´e modelov´anı vazk´eho nestlaˇciteln´eho proudˇenı profilovou mˇrıˇzı

103 Pages·2007·0.61 MB·Slovak

UNIVERZITA KARLOVA V PRAZE
´ ´
MATEMATICKO–FYZIKALNI FAKULTA
Katedra numericke´ matematiky
Matematicke´ modelova´n´ı vazke´ho
nestlacˇitelne´ho proudeˇn´ı
profilovou mrˇ´ızˇ´ı
ˇ ´ ´
DISERTACNI PRACE
ˇ
CERVEN 2007
RNDr. Toma´sˇ Neustupa
CHARLES UNIVERSITY, PRAGUE
FACULTY OF MATHEMATICS AND PHYSICS
Department of Numerical Mathematics
Mathematical Modelling of Viscous
Incompressible Flow through
a Cascade of Profiles
DISERTATION THESIS
JUNE 2007
RNDr. Toma´sˇ Neustupa
Acknowledgement
I would like to express my gratitude to my supervisor Prof. RNDr. Miloslav Feistauer,
DrSc.,Dr.H.C.forintroducingmetothefieldofincompressibleflowsthroughcascadesof
profiles, for encouragement, numerous discussions and hints and for his effort in guiding
andsupportingmystudies. Thedissertationwouldnothavebeenpossiblewithouthishelp.
I am grateful to the Faculty of Mathematics and Physics at the Charles University for
giving me the opportunity to spend and perform my doctoral studies at the Department of
NumericalMathematicsinapleasantandstimulatingenvironment.
I also thank Prof. RNDr. Karel Kozel, DrSc. for including me into the team of re-
searchers who work on the Research Plan of the Ministry of Education of the Czech Re-
public No. MSM 6840770010 at the Czech Technical University, Faculty of Mechanical
Engineering, Department of Technical Mathematics. It enables me to live and work in a
scientificcommunityandtocontinueinmyresearchafterfinishingmydoctoralstudies.
Andfinally,IwishtothankthechairoftheDepartmentofTechnicalMathematicsatthe
Czech Technical University, Faculty of Mechanical Engineering, Prof. Ing. Jaroslav Fort,
CSc. for supporting research generally and for creating very good working conditions at
thedepartmentinthelastyearswhenIamemployedthere.
1
Contents
Introduction ................................................................... 3
Geometryoftheproblem ....................................................... 6
I. Stationaryproblemwithanonlinearmixedboundaryconditionon
theoutflow
1. Classicalformulation ................................................... 9
2. Weakformulation ..................................................... 13
3. Existenceofaweaksolution ........................................... 30
4. Uniquenessoftheweaksolution ....................................... 41
II. Stationaryproblemwithmodifiedboundaryconditionsontheoutflow
1. Theproblemwithalinearmixedboundaryconditioninvolving
Bernoulli’spressureontheoutflow ..................................... 45
2. Theproblemwithlinearseparatedboundaryconditionsforvorticity
andBernoulli’spressureontheoutflow ................................. 51
3. Theproblemwithamodifiednonlinearboundaryconditiononthe
outflowandalargeinflow ............................................. 54
III. Nonstationaryproblemwithanonlinearmixedboundaryconditionon
theoutflow
1. Classicalformulation .................................................. 68
2. Weakformulation ..................................................... 70
3. Existenceofaweaksolution ........................................... 74
Appendix: Someimportantinequalities ....................................... 91
Listofsymbols ............................................................... 96
References ................................................................... 98
2
Introduction
Oneofthemostimportantsubjectsinthetheoryofblademachines(i.e. turbines,com-
pressors, pumps etc.) is the study of flows through blade rows. These flows are rather
complex. In general, they are three-dimensional, nonstationary, rotational, turbulent and
thedomainoccupiedbythefluidhasacomplicatedgeometry. Thisisthereasonthatthere
is a number of various models used for the analysis of the flow though blade rows. Sim-
ple one–dimensionalmodels whichgive only avery roughimage of thebehaviour of fluid
flowinbladerowsarenowemployedrarely. Ontheotherhand,three–dimensionalmodels
are so complex that their usage in numerical calculations requires rather long CPU time
even on most modern computers. This is why two–dimensional models are still exten-
sively used. We can mention, e.g. the widely used model of plane flow through a cascade
of profiles. (For its derivation, see e.g. [8] or [7], Section 3.4.) It has numerous variants
taking into account various features, as incompressibility or compressibility of the fluid
and variable geometry of casing walls of the through flow part of blade machines lead-
ing to the model of flow in a layer of variable thickness. (See, e.g. [2], [8], [29], [33],
[44], [52], [56], [57]). The model of plane irrotational incompressible inviscid flow, nu-
merically realized by Martensen’s method, was applied in engineering practice for several
decades. E.Martensentransformedtheproblemtoaboundaryintegralequationonasingle
profile of a cascade in his papers [45], [46] and [47]. Using the Fredholm alternative, he
proved the existence and uniqueness of the solution. Martensen’s integral equation was
later discretized and became the basis for an efficient and accurate numerical method for
thesolutionofflowsthroughcascadesofprofiles;see([31],[51]).
The method of E. Martensen can be considered to be the first mathematically founded
technique for the simulation of cascade flow problems. The development of modern com-
putersinthelastyearsenablesustoconsidersuccessivelymoreandmorecomplexmodels,
involving compressibility, viscosity, heat conduction and various other phenomena. A se-
ries of approaches to numerical simulation of cascade flow problems is described in ([3],
[4], [5], [6], [9], [10], [11], [12], [13], [14], [15], [17], [18], [19], [24], [26], [27], [28]).
However,arigorousmathematicaltheoryisinmostoftheusedmodelsmissing.
Motivated by this state, the present thesis deals with the theoretical analysis of an in-
compressible viscous flow through a plane cascade of profiles. Our model is based on the
Navier–Stokes equation (conservation of momentum) and the equation of continuity (con-
servationofmass). Inthefirstthreechapters,westudythestationaryproblemwithsuitable
boundary conditions. The conditions involve the non–homogeneous Dirichlet boundary
condition on the inflow, the no–slip boundary condition on the profiles and the condition
of periodicity in one space direction (along the cascade of profiles). The spatially peri-
odic character of the flows enables us to restrict ourselves to a boundary–value problem
formulatedjustononespaceperiod.
The mostly discussed condition is the boundary condition on the outflow. We use a
modified “do nothing condition” analogous to the condition used by J. Heywood, R. Ran-
nacherandS.Turek[25]inChapterI.Thename“donothing”comesfromthefactthatthe
condition does not explicitly appear in the weak formulation of the problem and it follows
from this formulation if an existing solution is in some sense “smooth”. If the convective
nonlineartermintheNavier–Stokesequationiswritteninthe“traditional”wayas(u·∇)u,
3
where u is the velocity, then the natural weak formulation (used in [25] and similar to the
weakformulationoftheboundaryvalueproblemwiththeDirichletboundaryconditionon
the whole boundary explained e.g. by R. Temam in [55]) does not exclude eventual back-
wardflowsontheoutputthatcanbringbacktotheconsideredflowfieldanuncontrollable
amount of kinetic energy and consequently, the energy estimate breaks down. The same
difficultywasalreadyobservedinstudiesofaflowinachannelbymanyauthors. Inorder
to preserve an energy estimate, S. Kracˇmar and J. Neustupa in [34] and [37] prescribed
an additional boundary condition which restricted the kinetic energy brought back on the
outflow and they have therefore described and solved the problem by means of a varia-
tional inequality of the Navier–Stokes type. P. Kucˇera and Z. Skala´k solved the problem
for “small” data, see [39], [38]. In Chapter I of this thesis, we use a certain nonlinear
modification of the boundary condition on the outflow from [25]. The modification was
proposed by C. H. Bruneau, F. Fabrie in [1]. It enables us to derive necessary energy es-
timates and to prove the existence (for a sufficiently small inflow) and uniqueness (for all
datasufficientlysmall).
In the first two sections of Chapter II, we use the Navier–Stokes equation with the non-
lineartermintheformcurlu×u(whosetwo–dimensionalanalogisω(u)u⊥ –seeSec-
tion II.1). This is compensated by the presence of pressure in Bernoulli’s dynamic form
p+1|u|2 (whichwecalltheBernoullipressure). Sinceω(u)u⊥ ispoint–wiseperpendicu-
2
lartou,weareabletoderive,fromacorrespondingweakformulation,anenergyestimate
regardless the backward flows on the outflow. Thus, the “do nothing” boundary condition
on the outflow, following from the weak formulation, need not be modified in the sense of
C.H.Bruneau,F.Fabrieandcanhaveformallythesameformasin[25],howeverwiththe
Bernoulli pressure q = p + 1|u|2 instead of the “static” pressure p. The energy estimate
2
further enables us to prove the existence and a uniqueness of the weak solution. We also
need the velocity on the inflow to be “small enough” in the proof of existence and all data
tobe“smallenough”inthetheoremonuniqueness.
Therequirementonthesmallnessofthedatainthetheoremsonuniquenesshasthesame
reason as the similar condition in known theorems on uniqueness of stationary solutions
of boundary–value problems for the Navier–Stokes equation with the Dirichlet boundary
condition on the whole boundary of the flow field. As examples of non–uniqueness are
known in the case of the Dirichlet boundary condition on the whole boundary and large
data, it is logical to expect that the non–uniqueness can also take place in the case of our
boundaryvalueproblem,ifthegivendataarelarge.
The condition on smallness of the inflow velocity has the same background as the con-
ditiononsmallnessoffluxesbetweenvariouscomponentsoftheboundaryinknowntheo-
remsonexistenceofaweaksolutionoftheboundary–valueproblemfortheNavier–Stokes
equation with the Dirichlet boundary condition, see e.g. G. P. Galdi [21] or V. Girault and
P.-A. Raviart [23]. The case of a large inflow is solved in Section II.3, however the pos-
sibility of the large inflow is again compensated by certain modification of the boundary
conditionontheoutflow.
ChapterIIIdealswiththenonstationarycaseoftheboundaryvalueproblemfromChap-
ter I. We present a weak formulation and we prove the existence of a weak solution.
Our result extends known theorems on the existence of a weak solution of the Navier–
Stokes initial–boundary value problem, proved originally by J. Leray [42] and E. Hopf
4
[30], and explained in a series of books on the theory of the Navier–Stokes equations,
see e.g. O. A. Ladyzhenskaya [41], V. Girault and P.-A. Raviart [23], R. Temam [55] and
M. Feistauer [7]. The proof is carried out by the Rothe method, which was for the first
timeappliedtotheNavier–StokesequationsbyM.Shinbrotin[54]. Duetothepossibility
of deriving a necessary energy estimate by means of the Gronwall lemma, we do not need
the velocity on the inflow to be “small” in Chapter III. On the other hand, the obtained
regularity of the weak solution does not enable us to prove its uniqueness even if all the
data are “small”. It can be expected that, in the case of “small” and regular data (inflow,
specific body force, outflow), the existing weak solution is regular and therefore unique,
as it is known from the case of the Dirichlet boundary condition on the whole boundary.
However,wehavenotstudiedthisquestioninthethesis.
5
Geometry of the problem
(cid:113)
(cid:113)
(cid:113)
(cid:113) etc.
(cid:113)
(cid:113)
(cid:54)
x
2
D
Γ2 : x = γ(x ) + 2τ
2 1
Ω
2
(cid:24)(cid:24)(cid:24)(cid:58) A2 C (cid:81)
1
(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:58)(cid:24)(cid:58) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) B (cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:115)(cid:81)(cid:115)
(cid:24)(cid:24)(cid:24)(cid:58) 2 (cid:81)(cid:81)(cid:81)(cid:115)
(cid:81)
(cid:24)(cid:24)(cid:24)(cid:58) Γ1 : x2 = γ(x1) + τ (cid:81)(cid:81)(cid:81)(cid:81)(cid:115)
(cid:24)(cid:24)(cid:24)(cid:58) A (cid:54) Ω1 (cid:81)(cid:81)(cid:81)(cid:81)(cid:115)
(cid:24)(cid:24)(cid:24)(cid:58) 1 τ C0 (cid:81)(cid:81)(cid:81)(cid:81)(cid:115)
(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:58)(cid:24)(cid:58) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) B1 (cid:81)x(cid:81)(cid:81)1(cid:81)(cid:81)(cid:81)(cid:115)(cid:81)(cid:115)(cid:45)
(cid:81)
(cid:24)(cid:24)(cid:24)(cid:58) Γ0 : x2 = γ(x1) (cid:81)(cid:81)(cid:81)(cid:115)
(cid:81)
(cid:24)(cid:24)(cid:24)(cid:58) A (cid:63) Ω0 (cid:81)(cid:81)(cid:81)(cid:81)(cid:115)
(cid:24)(cid:24)(cid:24)(cid:58) 0 C−1 (cid:81)(cid:81)(cid:81)(cid:115)
(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:58)(cid:24)(cid:58)(cid:24)(cid:58) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Ω−1 B0 (cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:115)(cid:81)(cid:115)
(cid:81)
(cid:24)(cid:24)(cid:24)(cid:58) (cid:81)(cid:81)(cid:81)(cid:115)
G G (cid:81)
i o (cid:81)
(cid:81)(cid:81)(cid:115)
(cid:113)
(cid:113)
(cid:113) etc.
x = d (cid:113) x = d
1 i (cid:113) 1 o
(cid:113)
Fig.1
LetussupposethatC isasimpleclosedcurveinR2 whichispiecewiseoftheclassC2
0
and whose interior and exterior are domains with a Lipschitz–continuous boundary. For
k ∈ Zweput
C = {(x ,x +kτ); (x ,x ) ∈ C },
k 1 2 1 2 0
6
whereτ isapositiveconstant. Weassumethatτ issolargethatthecurvesC aremutually
k
disjoint. ByIntC wedenotetheinteriorofthecurveC . Theset
k k
+∞
M = IntC
(cid:91) k
k=−∞
iscalledacascadeofprofilesandeachofitscomponentsIntC iscalledaprofile. Number
k
τ iscalledtheperiodofthecascade.
Let d and d be two real numbers which are chosen so that d < d and all the profiles
i o i o
are lying inside the strip {[x ,x ] ∈ R2; d < x < d }. We shall study the motion of a
1 2 i 1 o
viscousincompressiblefluidintheunboundeddomain
D = {[x ,x ] ∈ R2; d < x < d } \ M.
1 2 i 1 o
We suppose that the fluid enters the domain D through the straight line G , given by the
i
equation x = d , and it leaves the domain D through the straight line G , given by the
1 i o
equation x = d . Now it is logical to call the straight line G the inlet (or the inflow)
1 o i
and the straight line G the outlet (or the outflow). In accordance with the notation G and
o i
G , we further denote by G the union of all the curves C : G = ∪+∞ C . The set
o w k w k=−∞ k
G represents an infinite family of fixed walls inside the flow field; this is why we use the
w
subscriptw.
Let us assume the existence of a function x = γ(x ), defined in the interval [d ,d ],
2 1 i o
which belongs to C1([d ,d ]) and whose graph separates the profiles IntC and IntC .
i o 0 −1
It means that the curve C is lying above the graph of the function γ and the curve C
0 −1
is lying below the graph of γ. (See Fig. 1.) We denote by A = (d ,γ(d )) and B =
0 i i 0
(d ,γ(d ))theendpointsofthegraphofγ.
o o
Let us denote by Γk (for k ∈ Z) the curve which is identical with the graph of the
functionx = γ(x )+kτ ford ≤ x ≤ d .
2 1 i 1 o
Noweachofthesecurvesseparatestheprofiles IntC and IntC . WedenotebyA ,
k−1 k k
respectively B the end points of the curve Γk. It means that A = (d ,γ(d ) + kτ) and
k k i i
B = (d ,γ(d )+kτ).
k o o
LetusdenotebyΩ (fork ∈ Z)asubdomainofD,whoseboundaryconsistsoftheline
k
segmentsA A ,B B andthecurvesΓk,Γk+1 andC .
k k+1 k k+1 k
Our goal is to study flow in the whole domain D. This domain is unbounded, however
its shape repeats periodically with the period τ in the x –direction. Thus, it is reasonable
2
to deal with flow which is τ–periodic in the x –direction. This assumption enables us to
2
restrict our considerations the flow in one spatial period of the domain D only, e.g. the
flow in the domain Ω , naturally with appropriate conditions of periodicity on the curves
0
x = γ(x )andx = γ(x )+τ,d ≤ x ≤ d .
2 1 2 1 i 1 o
Letusfurther,forsimplicity,writeonlyΩinsteadofΩ andletusalsousethisnotation:
0
Γ – thelinesegmentA A ,
i 0 1
Γ – thelinesegmentB B ,
o 0 1
Γ – thegraphofthefunctionx = γ(x ),d ≤ x ≤ d ,
− 2 1 i 1 o
Γ – thegraphofthefunctionx = γ(x )+τ,d ≤ x ≤ d ,
+ 2 1 i 1 o
7
Γ – coincideswiththecurveC .
w 0
Thesymbols(Γ)◦,(Γ )◦,(Γ )◦,(Γ )◦ denotethecurvesΓ,Γ ,Γ ,Γ withouttheirend
i o − + i o − +
points:
(Γ)◦ – theopenlinesegmentA A ,
i 0 1
(Γ )◦ – theopenlinesegmentB B ,
o 0 1
(Γ )◦ – thegraphofthefunction x = γ(x ), d < x < d ,
− 2 1 i 1 o
(Γ )◦ – thegraphofthefunction x = γ(x )+τ, d < x < d .
+ 2 1 i 1 o
Usingthisnotation,wehave
∂Ω = Γ ∪ Γ ∪ Γ ∪ Γ ∪ Γ .
i o − + w
Fig.2showsthedomainΩwithallthepointsandcurveswhichformitsboundary.
(cid:54)
x
2
Γ
A (cid:54) +
1 Γ
τ w
(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:58)(cid:24)(cid:58) Γi (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) B
1 x
1 (cid:45)
(cid:81)
(cid:24)(cid:24)(cid:24)(cid:58) Γ (cid:81)(cid:81)(cid:81)(cid:115)
o (cid:81)
(cid:24)(cid:24)(cid:24)(cid:58) A (cid:63) Γ− Ω (cid:81)(cid:81)(cid:81)(cid:81)(cid:115)
0
(cid:81)
(cid:81)(cid:81)(cid:115)
(cid:81)
(cid:81)
(cid:81)(cid:81)(cid:115)
(cid:81)
B (cid:81)
0 (cid:81)(cid:81)(cid:115)
x = d x = d
1 i 1 o
Fig.2
8