An algorithm for solving the nonlinear stationary Navier-Stokes problem is developed. Explicit error estimates are given. This mathematical technique is potentially adaptable to the separation problem

Gabrielsen, R. E.

Karel, S.

English

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https://ntrs.nasa.gov/search.jsp?R=19760006340 2020-03-22T17:25:13+00:00Z
NASA TECHNICAL
MEMORANDUM
NASA TM X- 62,487
Steven Karel
U. S. Army Element, NASA
Washington, D. C. 20546
and
Ames Research Center
Moffett Field, Calif. 94035 ti	 I)EC 1975	 s `ti RECEIVED
NASA sn FAMUfY
INMIT Iwo
January 1975
i	 i •
(RASA-T!l-X-62497) A FAMILY OF APPROXIMATE
SOLUTIONS ANC EXPLICIR ER£OP ESTIMATES FOB
THE NONLINEAR STAT INARY 3AVIER-STOKES
X	 PROBLEN (NASA) 11 F EC $3.50	 CSCL 20C
JF
2
376-13428
Unclas
G3/34 05618
A FAMILY OF APPROXIMATE SOLUTIONS AND EXPLICIT ERROR ESTIMATES
FOR THE NONLINEAR STATIONARY NAVIER-STOKES PROBLEM
Ralph E. Gabrielsen
Ames Research Center
and
U. S. Army Air Mobility R&D Laboratory
Moffett Field, Calif. 94035
and
1. Report No.
NASA TM X-62,497
2. Goverment Accession No. 3. ReeipMrt'i coal" No.
4. Tkle and Subtiue
A FAMILY OF APPROXIMATE SOLUTIONS AND EXPLICIT
a Report Dees
& Fes forming OrprilasMon codeERROR ESTIMATES FOR THE NONLINEAR STATIONARY
NAVIER-STOKES PROBLEM
7. Author(s) & Mon ge OrpnWOm Report No.
Ralph E. Gabrielsent and Steven Kareltt A-6319
10. Work Unit No.
505-06-110. Par to wing oroenimkm No 0 and Address
*Ames Research Center and U.S. Army Air Mobility
R&D Laboratory, Moffett Field, Calif. 94035, and
tAmes Research Center, Moffett Field, Calif. 94035 and
11. Tech
ech
tni Grant
cal Memorandum
13. Type of Report and Period Cowed
12. Sponsoring Agency Nam and Address
National Aeronautics and Space Administration 14. Sponsoring Aomy Code
Washington, D. C. 20546
IS. supplementary, Nom
16. Abstract
An algorithm for solving the nonlinear stationary Navier-Stokes protlem
is developed.	 Explicit error estimates are given.
17. Key Wads (Suggaatad by Authorial) 15. Distribution Statement
Navier-Stokes equations (stationary)
Newton-Kantorovich method Unlimited
TAR CAtp-mv — 34
19. Security passes. (of this report) 20. Security Clpsif. (of this page) 21. No. of Pagss 22. Prfa'
Unclassified i	 Unclassifigd 10 $3.25
'For sale by the National Technical Information Sarvics, Springfield, Virginia 22151
A FAMILY OF APPROXIMATZ SOLUTIONS AND FSLIC?T ERROR ESTIMATES FOR THELfY'VLMIL'VLr ^L1 '	 'L.^.A4•L	 WL'
►-	
^-^•^^
NONLINEAR STATIONARY XAVIER-STOKES PROBLEM
-	
r443 ► /.L.vYL+i .. ♦4 ► .. ^ Y L 4 'L. ^.-.1...LL^6"L41..'L,-aA^s'+^.+
RALPH E. GABRIELSEN* AND STEVEN KARELt
Abstract. An algorithm for solving the nonlinear stationary Navier-Stokes
problem is developed. Explicit error estimates are given.
1. Introduction. Since the separation problem of aerodynamics is at
present intractable, it has been decided to undertake a closely related problem
via a mathematical technique that is potentially adaptable to the separation
problem. Specifically, the problem under consideration is the "nonlinear sta-
tionary Navier-Stokes problem" of fluid dynamics. The generalized Newton's
method, as developed by Kantorovich (5, 71 is used. Its application to this
pZlem is of definite value for those seeking practical solutions of related
fluid flow problems. The following questions are considered:
(i)Under what conditions does the sequence of funetiohs obtained by
Newton's method converge to the solution?
(ii)How should the initial guess be made, as a function of v, so as to
guarantee convergence?
(iii)At what rate does the sequence of approximate solutions converge?
Given S, a two-dimensional Green's domain, and f l (x,y) a Cl(S),
f2(x,y) a C I (S), the nonlinear stationary Navier-Stokes problem is:
*Ames Research Center and U.S. Army Air Mobility R&D Laboratory, Moffett
Field, Calif. 94035.
tU.S. Army Element, NASA, Washington, D.C. 20546 and Ames Research Center,
Moffett Field, Calif. 94035.
2uux +vY+ qx - v Qu+ fI(x,y) = 0
uvx +vy+ qy - v 4v+ f 2(x .Y ) = 0
with boundary conditions
UPS) _ —b2( aS) , v(aS) = b l (aS) .
An equivalent expression is
(2)
	
PW = v44* + 0v^x0 + f ly — f2x = 0
X(as) = b l (as)
	 V►v(aS) = b2(as) .
where
*x = v , V►Y = 
-u ,
and P is a mapping from C 4 (S) into C O (S) with norm
N n
N V►^^ = r
 anm x me-mCN
n= 0 m=0
	 ax ay
For clarity, the equivalence of (1) and (2) is now shown.
LEMM (1) '-' (2).
Proof.
(1) -► (2) directly follows from [6, Theorem 6, p. 131).
(2)-► (1): (2) can be readily rearranged into the form
(-v4u + uu + vu + f l ) _ (-v4v + uv + vv + f2)
x	 y	 y	 x	 y	 x
with
Let
Z=(-v4u+uux +vy+fl)i+(-v4v+uvx +vv+f2)I .
Note the fact: if V= ai + bj f C l (S), then
.} F + V = VF •-►
 Y = b 
3Hence, + q	 Vq.
Therefore, (2) + (1).
2. Main Results. We seek a Rnlutten to (2) by the generalised Newton's
method. Consider the equation
P 40) + P'(*o)(* - *0)	 0 .
If *1 is a solution of this equation, we can write a new equation
P(*0 + P= (W1)(W - V► 1)	 0 .
Assume that for each n 
> 0 + *n+1 }
P (On ) i` P'(*n) (*n+ 1 - *n ) - 0 '
(3)
Onx(as) a bl	 ny(as) - b2 .
(See LEMMA 1 for explicit expression for P'.)
If lim^n exists, let	 be the limit. Then
n—
P(**) = 0 .
Thus ** is the desired solution. This is Newton ' s method.
Define
N nEanG(x'.Y'.x.Y)
	= 	 n n-m
max ^	 ^Y
x'rY' E S	
n-0 m=0	
axe ayt
where G is the Green's function of the problem
eel =0 in s, SOS) =0, ,yn(as)=0.
(1], (2), (3], (41, (8).
Let M. 0 = max (max l 6*0 1, maxIt'*0 1,	 I^Ox 1, ^1*0Y .
Therefore, based on the remarkable theory developed by Kantorovich, we
obtain the following result:
THEOREM 1. If the initiaZ *0 is 4-
*0x 
las	 oYIBS
= bl, 	- b2, M *o H 3 < v s
and	
v2(l - V M*OH3)2
NP(* 0 )I 	 s	 2
C O 	2H4
then the Newton-Kantorovich sequence (*m)m=o does in fact converge to the
unique solution of (2).
Proof. By the theorem of Kantorovich 15, p. 7081, it is sufficient to
show that
IP(*0)I S	 1
21P I (Vo) —lr flP"I
this is shown by the following lemmas.
LEmm 1. P'(*), the Freehet derivative exists at aZZ points *0 in the
domain, and
P ' (*O) g = vAAg + % Agx + A*Oxgy - A*Oygx - * OxAgy .
Proof:
PW _ vav + *yA*x - *xA*y + f ly - f 2x .
If
lim IP(*+a) - P(*) - Lai = 0
Dal -0	
IaN
for some linear operator L, then define P'(* 0 ) = L.
P(*O+a) - P (*O) = vAAa + *OyAax + aA*0x + yAax - axA*ov - &Aay - axAay .
Let
L(*O) (a) = vAAa + *p r Aax + yA*Ox - axAfty - *oxA y .
Then L(^ O ) is a linear operator. Therefore, L(*) = P'(*) if
lim 
NP( * p+a) - P(*0) - Lal)= lim N yAax ax A : 0
1184,010
	
IaN
	 jaM-O	 MaN
aaax - yYeCO .
t^ I
5
Therefore,
a Aax - axA v^ max v
A x axA vS
a C C4 .
Therefore,	 4 n
	
H a l "	 ^max ana
n=0 m=0
l ax ayn-m
yaax
 - aXAar ^ 1 1 vI ( Ia^I + 
Ia ► 1) + IaxI (I Xxv ) + l a 	1) .
Therefore,
max Ia Aa-a && < max Ia 	 I+maxIa
S y	
x 
y	 S y S	 S '4'3'	 S	 S xxy S
So:
IP(4YO+a) - P(*O) - Lail
0 < lim
p al!-►O	 la)
5 lim
pap *0
max, JJ Sm , a	 I + max, a	 I + max Is, 1 a 	 I + max) 
a"- IS
	
` s	 s	 s	 `s	 s
a
-< lim NaN(Nall +Nall) + Map(NaN + Nall)
IaN,,O	 Nall
Therefore,
(1P(ft + a) - P (WO) - Lall
lim	 pa'	 = 0	 Q.,E.D.
I all-+O
LEMMA 2. P"(*) exists at aZZ points too in C `' ( S) and
P11 00 : YOX + ^YOX - Y*y - yly .
Proof. By definition, P"(^ O ) exists if there is a bilinear operator B
lim NP' 
(*0 + 0) - P' (*O) - B01 1
p 0 -*0
6If so, P"(VO) is defined to be B.
P ' ( *0+m )g - vaag + ( * 0+4)yagx + gya (*O+0X — gxa(* O+*)y — (*o+*) XagY .
[ P'(*o+#) — P'(*o)] g - ^yegx + gya^x — gxa^v — OXagy .
Now,
[P'(*o+o) - P'(*o )](g1+62) - [ P'(*O+O) - P '(*0) 191 + [P ' ( *O+* ) - P ' (*0)392 •
and
[P'(0 0+W1 2 ) - P'(*O)] g - [ P'Q-O+W - P'(*0)]g + [P'4002) - P'(*O)]g
Therefore, [P'(0 0+0) - P'(*O)] g is a bilinear operator of ^ and q. Let
BO = P'(* 0+¢) - P l (* O ). Then it follows that
P 11 40) 48 - rya@ X + eyox - OXA^Y - yeY .
IM44A 3. P' (*o ) -1 exists.
Proof. Given P(* O ) + P' ( *O)(*-Oo)	 0, let	 *0, then
P' 40)i - -P(*o) .
Equivalently, vaa^ + a*Oxjy + *Oya^x - *OxA^y - A*oy;X - -P(* () ). This equation
can be abbreviated as va4 - f(^) + F. Let G be the Green's function for
y (see Theorem 1), then
iy=1JG?+^ jGF.
Ato
Define the linear operators
A^ 1
v j (',f (V^ )
B[-P(* 0 )]
v
 ! GF
Then	
``
.
	 (I - 1 A); ' B[-P(*O)] .V
-1
Under proper conditions, as shown later in the proof, ^I - I Aa exists. Then
7{I - A)
-iBE -P( 0) ] •
Therefore,
i	 P' (00 ) tI - v Al iB [ -P (*0) ] _ -P(*p)
i	 J
and (I - A) B is a right inverse. Also, CI - A) iBP' ( *p)^
(I - A) IBP' ( gy p) - I, and (I - A) - 'B is a left inverse. Therefore,
``(I - A) i B = (P'(*O)]-i
i
We now show the conditions under Which CI - A, exists.
Any = 
1S 
G ( *X '*py - *3rA*ox + *oxA*'y - *oyl'*x)dS
OAII = sup NA*N
N^gsl
so
NAB = sup	 max j
S 
G(
*xA* Oy-V^yA* Ox+ ^ypxA*+ _*OyA*XMS I + max ( ) X I +	 .
C	 <1 1XI,y'eS^ 
IIAI
	
< sup	 max j[IGI + IGx11 + Y11 + . . .] • (1*x 11oOy 1 + . . .]
C	 AA S1 x ' •y ' Es
dAN	 S max	 max	 M f (IG I I IG , l + IG ^ I + .	 . ]	 t l^x I + I* 1 +
C	 1*1a 1 x' lY I CS * 0 	x	 y	 y
Therefore,
IIAII 4 = M*0 mm f t 1G1 + IGx ,1 
4 I Y' 1 + . .. ]dx dy .
NAI C4 s M*0H3 .
^m
Therefore, (I - A) i exists if M^oH 3 < v. ^I - v A) -i = j 1^  An exists.
n=0 
v
8LEMMA 4• NP"(*0)N -< 1.
.	 Proof.
max Io ao. + eynmx
 - ^xeey
 - 
yo.j
IAN l Y
JOIS1
loy l + 143= 1 + 14	 1+I#,1+I+,^,I ♦ lo,,,,y,l) s 1•
This completes the proof of Theorem 1.
THEOREM 2. Under the hypothesis of Theorem 1, the error estimate for the
mth approximate solution is expressed by
Hy
2^1-1
it 0 - ►mil
	
(22^m )
	
1	 l P (V^ 0) p 2m .
1(1 - 
v M* Hg^0
Proof. By the Kantorovich theory, this result follows from the hypothesis
of Theorem 1.
COROLLARY 1. As a function of v, for fixed m,
Q* - om il - 0 (vP+'+l)
COROLLARY 2. If the hypothesis of Theorem 1 is satisfied, then for a given
e > 0, there exists a denwrwrably infinite number of Zinear equations and solu-
tions Om as speoifled by (3) such that the entire famiZy of gym 's are within
the e-neighborhood of the exaot solution 0 of (2), i.e.,
M* - *JC4 < E .
Proof. Follow directly from Theorem 2.
i
I	 !	 t
9
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(1] I. BMAKA, K. REKTORYS, and F. VAICHLO, Alathematisehe EZastiaitatetheorLs
der B'bsnsn Pnobleme, Akademi-Verlag, Berlin, 1960.
[2] R. COURANT and D. HILBERT, Methods of AfathematioaZ Physics, Interscience,
New York, N.Y., 1953.
[33 P. R. GARABEDIAN, Partial DifferentiaZ Equatione, John Wiley, New York,
N.Y., 1964.
[4)  L. V. KATMROVICH and V. 1. KRYLOV, Approximate Methods of Higy.ar Analysis,
Interscience, The Netherlands, 1958.
[5] L. V. KANTOROVICH and G. P. AKILOV, Functional AnaZysis in Normed Spaces,
Pergamon Press, New York, N.Y., 1964.
(6] 0. A. LADYZHENSKAIvA, The Mathematioat Theory of Viscous Incompressible
FUVe, 2nd English ed., rev. and Engl. translated from the Russian by
Richard A. Silverman and John Chu, Gordon and Breach, New York, 1969.
[7] L. B. RALL, Computational Solution of NonZinear Operator Equations, John
Wiley, New York, N.Y., 1969.
[8] S. TIMOSHENKO and S. WOINOWSKY-KRIEGER, Theory of PZates and SheZle,
McGraw-Hill, New York, N.Y., 1959.


A family of approximate solutions and explicit error estimates for the nonlinear stationary Navier-Stokes problem

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