4 Transonic Flow Calculations by Least-Squares and Finite Element Methods 227com-In Sec.. 4 Transonic Flow Calculations by Least-Squares and Finite Element Methods 2314.7.1.. 4 Transonic
Trang 14 Transonic Flow Calculations by Least-Squares and Finite Element Methods 227
com-In Sec 4.6.3.2 we would like to discuss a method (due to M O Bristeau)which also makes use of an upwinding of the density; this method can be usedwith simplicial10 finite elements (triangles in two dimension, tetrahedra inthree dimensions) and has been very effective for computing flows at highMach numbers and around complicated two- and three-dimensional geo-metries
4.6.3.2 A modified discrete continuity equation by upwinding of the density in the
flow direction Following Jameson [l]-[4] and Bristeau [2], [3], we may write
the continuity equation (4.14) in a system of local coordinates {s, n}, where(see Fig 4.8) for a two-dimensional flow, s is the unit vector of the streamdirection (i.e., s = u/1 u | if u # 0) and n is the corresponding normal unit vector(conventionally oriented)
Using {s, n} and setting11
we obtain (from (4.14))
10 We use here the terminology of Ciarlet [1], [2].
1' U is the Mach number M*.
Trang 2228 VII Least-Squares Solution of Nonlinear Problemsthe elliptic-hyperbolic aspect of the problem is clear from (4.61) Actually(4.61) can also be written
[we have 1/2/ca = (y + l)/2].
EXERCISE 4.3 Prove (4.61), (4.62)
We use (4.62) to modify the discrete continuity equation (4.27) as follows:
Find 4>h e V h such that
(iii) U h , p h are upwinded approximations of U and p, respectively.
More precisely, we write the second integral in the left-hand side of (4.63) asfollows
= (-1) Z ^ Z ^ E m(T)hs(T)(U2h - 1 )+V ^ - V ^ - ^ , (4.64)
where
(a) l, h = {P^o i s the set of the vertices of ST h , with P o = T.E.;
(b) W; is the basis function of HI (cf (4.24)) associated with P t by
w, e Hi Vi = 0, ,N h , Wi (Pd = 1, WiiPj) = 0, \/jjt i; (4.65) (c) ^ is the subset of 2T h consisting of those triangles having P i as a commonvertex;
(d) m(T) = meas(T);
Trang 34 Transonic Flow Calculations by Least-Squares and Finite Element Methods
Figure 4.9
229
(e) {iiji}j= i is an approximation of \(/)J| \<f> h | at vertex P; obtained, for
j = 1, 2, by the following averaging formulas:
I
'50»
(others averaging methods are possible);
(f) h s (T) is the length of T in the s direction, i.e.,
where wfcr, k = 1, 2, 3 are the basis functions associated with the vertices
P kT of T; actually (4.67) can also be written
(4.68)
(g) We define C/j as follows:
With each vertex P t o{ST h we associate an inflow triangle T { which is the
triangle of ?7~ h having P t as vertex and which is crossed by the vector
u; — i u ij}j=I pointing to P t (as shown on Fig 4.9); we then define Uf as
Uf =
cl
Trang 4230 VII Least-Squares Solution of Nonlinear Problems
and for each triangle T of 3~ h ,
T
T
(4.69)
(h) We finally obtain p h e Hi as follows:
As for Uf, we define p t by
Pi = P(<t>h)\Ti,
and then
4.6.3.3 Some brief comments on the least-squares sohrtion of (4.63) For solving
the discrete upwinded continuity equation (4.63), we can use the following squares formulation:
algorithm (4.33)-(4.40) with J h ( •) denned as in (4.71); however, more iterations
are needed)
4.7 Numerical experiments
In this section we shall present some of the numerical results obtained using theabove methods The results of Sec 4.7.1 are related to a NACA 0012 airfoil;those of Sec 4.7.2 (resp., 4.7.3) to a Korn airfoil (resp., a two-piece airfoil)
Trang 54 Transonic Flow Calculations by Least-Squares and Finite Element Methods 231
4.7.1 Simulation of flows around a NACA 0012 airfoil.
As a first example we have considered flows around a NACA 0012 airfoil atvarious angles of attack and Mach numbers at infinity The correspondingpressure distributions on the skin of the airfoil are shown on Figs 4.10-4.17
in which the isomach lines (in the supersonic region only on Figs 4.10-4.14) are
also shown The results shown in Figs 4.10-4.14 have been obtained using theinterior penalty method of Sec 4.6.2; those of Fig 4.15-4.17 have been obtainedusing the upwinding method of Sec 4.6.3
We observe that the physical shocks are quite neat and also that the transition(without shock) from the subsonic to the supersonic region is smoothlyrestituted, implying that the entropy condition has been satisfied The abovenumerical results are very close to those obtained by various authors usingfinite difference methods (see, particularly, Jameson [1])
4.7.2 Flow around a Horn's airfoil
In Fig 4.18 we have represented the pressure distribution corresponding to theflow around a Korn's airfoil atM0O = 0.75 and a — 0.11; the computation
method is the interior penalty method of Sec 4.6.2 The agreement with afinite difference solution is good, as indicated in Fig 4.18
4.7.3 Flows around a two-piece airfoil
The tested two-piece airfoil is shown on Figs 4.19 and 4.20 Each piece is aNACA 0012 airfoil (the body No 1 is the upper body) The pressure distributionand the isomach lines (computed by the interior penalty method of Sec 4.6.2)are shown on Figs 4.19 and 4.20 We observe that the region between the twoairfoils acts as a nozzle; we also observe supersonic regions, in particular,between the two airfoils
4.8 Transonic flow simulations on large bounded computational domains
Consider the situation depicted in Fig 4.3; if the supersonic zone extends farfrom the airfoil, it is necessary to use a very large computational domain.Let 0 be the origin of coordinates; it is then reasonable to take the circle
{x eU 2 ,r = R x } for r ^ , with r = y/x{ + xf Now suppose that the disk of
center 0 and radius Ro is sufficiently large to contain the airfoil B in its interior;
we then introduce
Qi = {x e U2 , 0 < r < R o , x $ B},
Q2 = {xeR2,J?0 <r<RJ.
Trang 8234 VII Least-Squares Solution of Nonlinear Problems
B o
<
U
<
z
Trang 94 Transonic Flow Calculations by Least-Squares and Finite Element Methods 235
-II
•g
<
Trang 134 Transonic Flow Calculations by Least-Squares and Finite Element Methods 239
- 1
Figure 4.17 NACA 0012 airfoil M = 0.9; a = 0 (Computation with upwinding of the density.)
Trang 174 Transonic Flow Calculations by Least-Squares and Finite Element Methods 243
With {r, 9} as a standard polar coordinate system associated with 0, we define
the following new variables:
— X 2 , V X — {Xl 5 X;
(4.73)
Z 2 = 6, V{r,0}eQ2;
we use the notation £, = {^1; £2} in the sequel
We now introduce Qt = £l u and
Trang 18244 VII Least-Squares Solution of Nonlinear Problemsand
(4.76)The discretization of (4.75), (4.76) can be done as follows: On the one hand, one
uses in Q.1 (which is part of the physical space) a standard finite element
ap-proximation taking into account the possible complexities of the geometry; onthe other hand, since fi2 is a rectangle, one may use a finite difference discretiza-tion which can in fact be obtained via a finite element approximation on auniform triangle or quadrilateral grid (like the one in Fig 3.1 of Sec 3.5.4 of thischapter)
Remark 4.9 From the above decomposition of the computational domain
(in Cll and Q2), it is natural to solve the approximate problems by iterativemethods, taking this decomposition into account, and also the fact that finitedifferences can be used in Q2 (since finite differences allow special solvers on Q2)
We are presently working on such methods and also on their generalization tothree-dimensional problems; the corresponding results will be presented in aforthcoming paper
5 Numerical Solution of the Navier-Stokes Equations for
Incompressible Viscous Fluids by Least-Squares and
Finite Element Methods
5.1 Introduction Synopsis
The numerical solution of the Navier-Stokes equations for incompressibleviscous fluids has motivated so many authors that giving a complete biblio-graphy has become an impossible task Therefore, restricting our attention tovery recent contributions making use of finite element approximations, weshall mention, among many others, B G 4P [1], Bristeau, Glowinski, Periaux,Perrier, and Pironneau [1], Bristeau, Glowinski, Mantel, Periaux, Perrier, andPironneau [1], Glowinski, Mantel, Periaux, Perrier, and Pironneau [1],Gartling and Becker [1], [2], Hughes, Liu, and Brooks [1], Temam [1],Bercovier and Engelman [1], Fortin and Thomasset [1], Girault and Raviart[1], Le Tallec [2], [3], Johnson [2], Glowinski and Pironneau [1], [2],Gresho, Lee, Chan, and Sani [1], Rannacher [1], Benque, Ibler, Keramsi, andLabadie [1], Thomasset [1], and Brooks and Hughes [1]; see also the referencestherein
Trang 195 Numerical Solution of the Navier-Stokes Equations 245
In this section (which very closely follows Bristeau, Glowinski, Mantel,Periaux, Perrier, and Pironneau [1]) we would like to discuss several methods
for the effective solution of the above Navier-Stokes problems in the steady 12
and unsteady 13 cases The basic ingredients of the methods to be describedare the following:
(i) Mixed finite element approximations of a pressure-velocity formulation
of the original problem
(ii) Time discretizations of the unsteady problem by finite differences;
several schemes will be presented
(iii) Iterative solution of the approximate problems by least-squares
con-jugate gradient methods (possibly combined with an direction method)
alternating-(iv) Efficient solvers for the discrete Stokes problems.
The possibilities of the above methodology will be illustrated by the results
of various numerical experiments concerning nontrivial two-dimensional flows
5.2 Formulation of the steady and unsteady Navier-Stokes equations for incompressible viscous fluids
Let us consider a Newtonian, viscous, and incompressible fluid If Q and Fdenote the region of the flow14 and its boundary, respectively, then this flow
is governed by the Navier-Stokes equations
(i) u = {wJfL j is the flow velocity,
(ii) p is the pressure,
(iii) v is the viscosity of the fluid (v = I/Re, where Re is the Reynold'snumber),
(iv) f is the density of external forces;
12 One also says stationary.
13 One also says nonstationary.
Q a W, N = 2, 3 in practice.
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in (5.1), (5.3), (u • V)u is a symbolic notation for the nonlinear (vector) term:
Boundary conditions have to be added; for example, in the case of the airfoil,
B of Fig 4.1 of Sec 4.3.2 of this chapter, we have (since the fluid is viscous) thefollowing adherence condition:
u(x, 0) = uo(x) a.e on Q, (5.7)where u0 is given, is usually prescribed
Other boundary and/or initial conditions may be prescribed (periodicity inspace and/or time, pressure given on 3Q or on a part of it, etc.)
In two dimensions it may be convenient to formulate the Navier-Stokes
equations using a stream-function-vorticity formulation (see, e.g., Bristeau,
Glowinski, Periaux, Perrier, and Pironneau [1, Sec 4], Fortin and Thomasset[1], Girault and Raviart [1], Glowinski and Pironneau [1], Reinhart [1], andGlowinski, Keller, and Reinhart [1]) To conclude this section, let us mentionthat a mathematical analysis of the Navier-Stokes equations for incompressibleviscous fluids can be found in, e.g., Lions [1], Ladyshenskaya [1], Temam [1],and Tartar [1]
5.3 A mixed finite element method for the Stokes and Navier-Stokes problems
5.3.1 Synopsis
In this section we discuss a mixed finite element approximation of the Stokes problems which have been introduced in Sec 5.2 For simplicity weshall begin our discussion with the approximation of the steady Stokes problemfor incompressible viscous fluids, i.e.,
Navier vAu + \p = f in Q,
V • u = 0 in Q; (5-8)
as boundary conditions, we choose
u = g on T (5.9)(with j r g • n dF = 0, n being the unit vector of the outward normal at F), more
general boundary conditions are discussed in Appendix 3
Trang 215 Numerical Solution of the Navier-Stokes Equations 247
Also, for simplicity, in the following we suppose that Q is a bounded polygonal domain of U 2; but the following methods are easily extended to domains with
curved boundary in U 2 and U 3
5.3.2 A mixed variational formulation of the Stokes problem (5.8), (5.9) 5.3.2.1 Some functional spaces: Standard formulation of the Stokes problem.
The following (Sobolev) spaces play an important role in the sequel (in factthey have been extensively used in the other parts of this book):
sH\Cl), ^ L e L 2(Q), V i,j, 1 < i,j <
with
9(0.) = {(p e C^iU), 4> has a compact support in Q}.
From these spaces we also define
V g = {y\ye{H\Q)f, V • v = 0 in Q, v = g on T}.
Suppose that f e(H~\Q)) N , where H~\Q) = (HU(Q))* = dual space of Hj(Q)
and that g e (H V2 {T)f, j r g • n dY = 0; it then follows, from Lions, Temam, Ladyshenskaya, Tartar, loc cit., that (5.8), (5.9) has a unique solution {u, p} e V g x (L 2 (Q.)/U) If fe(L 2 (Q)) N and ge(H 3 ' 2 (T)) N , then {u, p} e (V g n (H 2 (Q)) N ) x (H^nyU) if T is sufficiently smooth The above u is also the unique solution of the following variational problem (where V o is obtained
by setting g = 0 in the above definition of V g ):
Find ueV n such that
Trang 22248 VII Least-Squares Solution of Nonlinear Problems
We now consider the following variational problem15
Find {n,\jj} eW g such that
v \vu-S\dx = f f - 0 + S<t>)dx, V{v, cj)}eW 0 (P)
In Glowinski and Pironneau [2] the following has been proved:
Theorem 5.1 Problem (P) has a unique solution {u, i//} such that
* = 0, (5.12)
u is the solution of the Stokes problem (5.8), (5.9) (and (5.10)) (5.13)
EXERCISE 5.1 Prove Theorem 5.1
Remark 5.1 The potential </> introduced above is not at all mysterious Indeed,
the formulation (P) can be interpreted as follows: if v e (H 1 (Q)) N and if F is
sufficiently smooth, there exist 0 e H 2 (Q.) n HQ(Q) and to e (H 1 (Q)) N with
V • o) = 0, such that
v = -\cj) + co, (5.14)
and the decomposition (5.14) is unique
In the formulation (P), instead of directly imposing V • v = 0, we try to
impose <f> = 0; these procedures are equivalent in the continuous case but not
at all in the discrete case, as will be seen below
5.3.3 A mixed finite element approximation of the steady Stokes
problem (5.8), (5.9)
As mentioned before, Q is a bounded polygonal domain of IR2 In this section wefollow B.G 4P [1] and Glowinski and Pironneau [2], [3]
5.3.3.1 Triangulation of Q.: fundamental discrete spaces Let {$~ h } h be a family
of triangulations of Q such that Q = [J Terh T We set h(T) = length of the greatest side of T, h = maxr s 5-h h(T), and we suppose that
Trang 235 Numerical Solution of the Navier-Stokes Equations
Figure 5.1
249
(where, in (5.19), g h is a convenient approximation of g whose construction will
be discussed in Appendix 3; if g = 0, one takes g h = 0 in (5.19) to obtain V Oh ),
Wgh = j { v , , 4>h} e Vgh x Hhh, , dx = • v hwh dx, V wh e
(5.20)
We shall also use the variants of V gh , W gh obtained from
V h = {v, e (C°(H))2, vh | r e P , x P:, V T e # , } , (5.21)
where #f t is the triangulation obtained from T h by subdividing each triangle
T e 2T h into four subtriangles (by joining the mid-sides; see Figure 5.1) In the
above definitions, P k denotes (as usual) the space of the polynomials in two
In Glowinski and Pironneau [2] one may find the proof of the following:
Theorem 5.2 Problem (P h ) has a unique solution {u h , ^/ h ) which is characterized
by the existence of a discrete pressure p h e H\ such that
{\Ph-\whdx= [f-\whdx, Vw.ef
v f Vu, • Vv, dx = f ( - \p h + f) • v, dx, V
•In Jn
{u h , i// h } E W gh (with >j/ h *Oin general).
EXERCISE 5.2 Prove Theorem 5.2