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Theory of three-dimensional laminar boundary layers
TREORY OF THREE-DIMENSIONAL LAMINAR BOUNDARY LAYERS* Yu.N. ERMAK and V.Ya. NEILAND (MOSCOW) (Received
3
January
1964)
Accurate methods for the numerical calculation (see [d, [A) of plane and axisymmetrical boundary layers have now been developed. However, although many theoretical examinations have been made in recent years (see [31 - [s]) of the boundary layer in three-dimensional flow, there is no sufficiently universal method which makes it possible to obtain accurate numerical solutions for a three-dimensional boundary layer. Even the approximate methods (see [51) involve considerable difficulties. In the three-dimensional problem, potential flow beyond the limits of the boundary layer depends on two coordinates of the body surface and the flow rate in the boundary layer has all the three components which in the most general case depend on all the three coordinates. Numerous effects are observed in a three-dimensional boundary layer which based on the two-dimensional theory, cannot be explained even qualitatively (e. g. secondary flows, divergence of flow lines etc. ). The three-dimensional boundary layers formed by the asymmetrical flow of blunt bodies and blunt front edges of wings and by the flow of triare of considerable practical interangular wings at angles of attack, est. Equations for a three-dimensional boundary layer in longitudinal orthogonal coordinates take the form
l
2%. Vych. Mat. 4. NO. 5. 950-954, 230
1964.
Three-dimensional
where the length We express tions (1)
laminar
boundary
component ds2 = dz2 + dhTdx2 + hidy2,
the boundary conditions
governing
u = u = w = OandH = H, u-
Ul (5
u ---c v, 66 Y),
Y),
If we assume Pr = 1, the energy (1) under certain conditions.
A,
layers
a H=C,T
+ $ (u2 +
the differential
v2i.
eaua-
with z = 0, (I, IJ) with z -
H-HI
integral
231
can be obtained
00.
from equations
In fact, we multiply the first equation of system (1) by a constant second by a constant B and add them up; as a result, if we assume
the
~P~~~_A~~_B~~+B~~+~~=O,
(2)
we obtain H=Au+Bv+C.
(3)
Thus, we have obtained a generalization of the Crocco integral for a three-dimensional boundary layer. Unlike the case of two-dimensional flow, when the Crocco integral is obtained for an external flow without gradient, integral (3) exists even in the presence of speed gradients at the external limit of the boundary layer. Indeed, let h, = A, (r) and h, = Ax(y). Then, P = P (z43[e, -
where
BR,),
x
I/
SLe,= s h, dz,
XI
In particular,
with
Ye,=Sh,dy. YI
A, = A, = 1 (on a cylinder) P = P (Ar - By).
Integral (3) applies also to flow free from gradient, if P = const. It is necessary to note that for “frozen” flow with Le = 1 and Pr = 1 integrals of the diffusion equations can be obtained by the same method, with the same assumptions concerning pressure ci = niu $ EiV _t F,. As a result of the success in obtaining accurate methods of calculation in relation to a two-dimensional boundary layer, it became possible to
solve
refers tro-
the to
problem
those
cases
and one-dimensional.
of
the
three-dimensional
when the
boundary
three-dimensional
In fact,
on flow
lines
problem
layer. is
This reduced
in the region
to
of the
Yu.N.
232
Ermak
and
V.Ya.
Nciland
critical point, the problem essentially becomes one-dimensional or twodimensional for which satisfactory methods of solution have been developed. We will illustrate this by several examples. We will consider asymmetrical flow in the region of the critical point. System (1) in this case is reduced to the system of ordinary differential equations
(NFJ’ + FoFif g - (F;)a= - c&F,“,
c[~-(GJg=-FoG,,
(NC;;)’+ cc,c; +
, g h;, + i 1
(F, + CC,) h;
=
0,
where N = pp/p~~~, and c = a@ (a and @ are coefficients external limit lJ N ax; V z By) with boundary conditions F = F’ = G = G’ = 0, F’ = G’ = 1, h=l
of
speed on the
h = O,with< = 0; with
(5)
c ---rm.
System (4) was solved by a method described in [?I. The thermodynamic and transfer properties of a gas at high temperatures are as given in [81. [91. The results are tabulated. A simple formula was calculated for engineering applications
f%a
-Gq (plr) (0.25 c + 0.75). U
B = 0.778 Pros4 V- Re,
It should be mentioned that a similar problem for incompressible liquid has been solved in [81 without the energy equation. We will now show how the three-dimensional boundary layer equations appear on the flow line of a cylindrical body (A1= hp = 1). For simplisystem we consider incompressible flow (Pr = 1). The coordinate city, will be selected as follows: coordinate z is directed along the normal to the body surface. y is the longitudinal coordfnate on the body surface and x is along the normal to the (y, z) plane. If form
the speeds on the external .!.I= x [-
the functions
f’ (y) -p al, F
V =
and G related
can oe introduced,
where
limit
f (y),
of
then
to flow
the boundary layer
take the
&Y/&w+ aVli+y + aWl& = 0
and
functions
E = zlL, q = y/L
and
c = zvIc/L
(L
is
the
8000
8000
10
t
273 0.00651
0.0564:
273 0.0072:
p=const, Tw =o
0.0018000
0.1207
“W
273 0.01081
2i3
0.0018000 mO0
2000
T,
10
P
”
h(o)
gjo,
(0,
40)
r$o,
‘w
kF!O)
(0,
60)
h;o,
i h(o)
I
( tp,
i
ry
/iOF
1.31 1.31 0.683
0.366
0.411
0.552 0.552 0.361
0.586 0.586 0.451
1.05 1.05 0.5227
1
1.30 1.25 0.667
0.578 0.564 0.440
0.9
1.29 l.i9 0.648
0.331
0.396
0.532 0.505 0.351
0.569 0.542 0.428
1.03 0.96 0.496
0.8
1.28 1.13 0.638
0.56i 0.518 0.4i7
0.7
i.0
1.27 1.06 0.612
0.5i7 0.460 0.338
0.553 0.493 0.405
0.86 0.471
0.0
1.26 0.9981 0.594
0.310
0.364
0.544 0.466 0.393
0.5
e
1.259 0.925 0.577
0.502 0.410 0.321
0.536 0.438 0.380
0.747 0.445
0.988
0.4
1.25 0.648 0.562
0.437 0.377 0.308
0.527 0.408 0.366
0.3
1.24 0.764 0.542
0.334
0.479 0.347 0.299
0.518 0.374 0.356
0.965 0.61 0.418
0.2
1.238 1.675 0.527
1.232 0.570 0.509
0.270
0.308
0.466 0.276 0.280 0.472 0.314 0.289
0.437 0.397
0.948
0
0.514 0.306 0.336
0.1
0.510 0.337 0.345
--
234
Yu.N.
Ermak and V. Ya.
and
Neil
characteristic dimension). with an accuracy of up to
Then. system (1) in the vicinity the second term of the expansion
tions
5) and h(<,
with
F(c,
q,
5).
boundary
CCC, q,
q,
of < = 0 of the func-
5) in < can be changed
to
conditions F,, =
C,, =
F; =
G; =
0,
h =
h,,
6 =
0, (7)
(*,
is
the
Thus,
characteristic
the
problem
speed); is
to
Unknown functions Fo, Go, ables (q. 5) so that this
solve
g = a/Vc. system
and hc depend
system
can
(6)
with
only
boundary
conditions
on two independent
be solved
(7).
vari-
by known methods. Translated
by
E. Semere
REFERSNCES
1.
Dorodaitsya.
A.A..
Prikl.
rckh.
Tekhn.
Fir.
9-10,
No.3.
111-116.
1960. Nauk SSSR.,
Petukhov.
3.
Mur. F., Theory of three-dimensional aeraogo pogranicbnogo sloya). mekhaniki’), Vol. II. Moscor.
4.
I. V. , Dokl.
Akad.
2.
132. boundary
layers
Sump. Mechanical Ixd-vo in. lit.,
T., Similar Boundary Layers on Solids pogranichaye 8101 aa telakh vrashcheniya).
Gauss,
No. 2.
of
367-310.
1960.
(Teoriya trekh(‘Problemy
problems
239-296. Revolution
Symp.
Problcna
1959. (Podobaye
Three-dimensional
pogranichnogo
sloya
Gosenergoizdat, 5.
i
laninar
voprosy
248-256.
Timman. R. and Tsaat.
boundary
235
layers
MOSCOW -
teploperedachi.
Leningrad,
1960.
Y&A_. , klcthods
of
calculation
related
to
three-
(Xetody rascheta trekhmernogo laminarnogo pogranichnogo sloya). Symp. Problema pogranichnogo s loya i voprosy tep lopercdachi. Gosenergoizdat, MOSCOW - Leningrad, 360-371, 1960. dimensional
laminar
6.
Howarth. L.,
7.
Neiland,
8.
Physical
9.
Tables
Phi 1.
V.Ya.,
boundary
Msg.,
Inch.
42,
Zh.,
layers
No. 1,
2,
gas dynamics (Fizicheskaya Nauk SSSR, Moscow, 1959. of
thernodynanic
kikh funktsii
1433-1441,
31-36,
1951.
1961.
gazovaya
dinamika)
Izd-vo
Akad.
of air (Tablitsy termodinamicbesVTs Akad. Nauk SSSR, Moscow, 1962.
functions
vozdukha),
335.
3
January
1964)
Accurate methods for the numerical calculation (see [d, [A) of plane and axisymmetrical boundary layers have now been developed. However, although many theoretical examinations have been made in recent years (see [31 - [s]) of the boundary layer in three-dimensional flow, there is no sufficiently universal method which makes it possible to obtain accurate numerical solutions for a three-dimensional boundary layer. Even the approximate methods (see [51) involve considerable difficulties. In the three-dimensional problem, potential flow beyond the limits of the boundary layer depends on two coordinates of the body surface and the flow rate in the boundary layer has all the three components which in the most general case depend on all the three coordinates. Numerous effects are observed in a three-dimensional boundary layer which based on the two-dimensional theory, cannot be explained even qualitatively (e. g. secondary flows, divergence of flow lines etc. ). The three-dimensional boundary layers formed by the asymmetrical flow of blunt bodies and blunt front edges of wings and by the flow of triare of considerable practical interangular wings at angles of attack, est. Equations for a three-dimensional boundary layer in longitudinal orthogonal coordinates take the form
l
2%. Vych. Mat. 4. NO. 5. 950-954, 230
1964.
Three-dimensional
where the length We express tions (1)
laminar
boundary
component ds2 = dz2 + dhTdx2 + hidy2,
the boundary conditions
governing
u = u = w = OandH = H, u-
Ul (5
u ---c v, 66 Y),
Y),
If we assume Pr = 1, the energy (1) under certain conditions.
A,
layers
a H=C,T
+ $ (u2 +
the differential
v2i.
eaua-
with z = 0, (I, IJ) with z -
H-HI
integral
231
can be obtained
00.
from equations
In fact, we multiply the first equation of system (1) by a constant second by a constant B and add them up; as a result, if we assume
the
~P~~~_A~~_B~~+B~~+~~=O,
(2)
we obtain H=Au+Bv+C.
(3)
Thus, we have obtained a generalization of the Crocco integral for a three-dimensional boundary layer. Unlike the case of two-dimensional flow, when the Crocco integral is obtained for an external flow without gradient, integral (3) exists even in the presence of speed gradients at the external limit of the boundary layer. Indeed, let h, = A, (r) and h, = Ax(y). Then, P = P (z43[e, -
where
BR,),
x
I/
SLe,= s h, dz,
XI
In particular,
with
Ye,=Sh,dy. YI
A, = A, = 1 (on a cylinder) P = P (Ar - By).
Integral (3) applies also to flow free from gradient, if P = const. It is necessary to note that for “frozen” flow with Le = 1 and Pr = 1 integrals of the diffusion equations can be obtained by the same method, with the same assumptions concerning pressure ci = niu $ EiV _t F,. As a result of the success in obtaining accurate methods of calculation in relation to a two-dimensional boundary layer, it became possible to
solve
refers tro-
the to
problem
those
cases
and one-dimensional.
of
the
three-dimensional
when the
boundary
three-dimensional
In fact,
on flow
lines
problem
layer. is
This reduced
in the region
to
of the
Yu.N.
232
Ermak
and
V.Ya.
Nciland
critical point, the problem essentially becomes one-dimensional or twodimensional for which satisfactory methods of solution have been developed. We will illustrate this by several examples. We will consider asymmetrical flow in the region of the critical point. System (1) in this case is reduced to the system of ordinary differential equations
(NFJ’ + FoFif g - (F;)a= - c&F,“,
c[~-(GJg=-FoG,,
(NC;;)’+ cc,c; +
, g h;, + i 1
(F, + CC,) h;
=
0,
where N = pp/p~~~, and c = a@ (a and @ are coefficients external limit lJ N ax; V z By) with boundary conditions F = F’ = G = G’ = 0, F’ = G’ = 1, h=l
of
speed on the
h = O,with< = 0; with
(5)
c ---rm.
System (4) was solved by a method described in [?I. The thermodynamic and transfer properties of a gas at high temperatures are as given in [81. [91. The results are tabulated. A simple formula was calculated for engineering applications
f%a
-Gq (plr) (0.25 c + 0.75). U
B = 0.778 Pros4 V- Re,
It should be mentioned that a similar problem for incompressible liquid has been solved in [81 without the energy equation. We will now show how the three-dimensional boundary layer equations appear on the flow line of a cylindrical body (A1= hp = 1). For simplisystem we consider incompressible flow (Pr = 1). The coordinate city, will be selected as follows: coordinate z is directed along the normal to the body surface. y is the longitudinal coordfnate on the body surface and x is along the normal to the (y, z) plane. If form
the speeds on the external .!.I= x [-
the functions
f’ (y) -p al, F
V =
and G related
can oe introduced,
where
limit
f (y),
of
then
to flow
the boundary layer
take the
&Y/&w+ aVli+y + aWl& = 0
and
functions
E = zlL, q = y/L
and
c = zvIc/L
(L
is
the
8000
8000
10
t
273 0.00651
0.0564:
273 0.0072:
p=const, Tw =o
0.0018000
0.1207
“W
273 0.01081
2i3
0.0018000 mO0
2000
T,
10
P
”
h(o)
gjo,
(0,
40)
r$o,
‘w
kF!O)
(0,
60)
h;o,
i h(o)
I
( tp,
i
ry
/iOF
1.31 1.31 0.683
0.366
0.411
0.552 0.552 0.361
0.586 0.586 0.451
1.05 1.05 0.5227
1
1.30 1.25 0.667
0.578 0.564 0.440
0.9
1.29 l.i9 0.648
0.331
0.396
0.532 0.505 0.351
0.569 0.542 0.428
1.03 0.96 0.496
0.8
1.28 1.13 0.638
0.56i 0.518 0.4i7
0.7
i.0
1.27 1.06 0.612
0.5i7 0.460 0.338
0.553 0.493 0.405
0.86 0.471
0.0
1.26 0.9981 0.594
0.310
0.364
0.544 0.466 0.393
0.5
e
1.259 0.925 0.577
0.502 0.410 0.321
0.536 0.438 0.380
0.747 0.445
0.988
0.4
1.25 0.648 0.562
0.437 0.377 0.308
0.527 0.408 0.366
0.3
1.24 0.764 0.542
0.334
0.479 0.347 0.299
0.518 0.374 0.356
0.965 0.61 0.418
0.2
1.238 1.675 0.527
1.232 0.570 0.509
0.270
0.308
0.466 0.276 0.280 0.472 0.314 0.289
0.437 0.397
0.948
0
0.514 0.306 0.336
0.1
0.510 0.337 0.345
--
234
Yu.N.
Ermak and V. Ya.
and
Neil
characteristic dimension). with an accuracy of up to
Then. system (1) in the vicinity the second term of the expansion
tions
5) and h(<,
with
F(c,
q,
5).
boundary
CCC, q,
q,
of < = 0 of the func-
5) in < can be changed
to
conditions F,, =
C,, =
F; =
G; =
0,
h =
h,,
6 =
0, (7)
(*,
is
the
Thus,
characteristic
the
problem
speed); is
to
Unknown functions Fo, Go, ables (q. 5) so that this
solve
g = a/Vc. system
and hc depend
system
can
(6)
with
only
boundary
conditions
on two independent
be solved
(7).
vari-
by known methods. Translated
by
E. Semere
REFERSNCES
1.
Dorodaitsya.
A.A..
Prikl.
rckh.
Tekhn.
Fir.
9-10,
No.3.
111-116.
1960. Nauk SSSR.,
Petukhov.
3.
Mur. F., Theory of three-dimensional aeraogo pogranicbnogo sloya). mekhaniki’), Vol. II. Moscor.
4.
I. V. , Dokl.
Akad.
2.
132. boundary
layers
Sump. Mechanical Ixd-vo in. lit.,
T., Similar Boundary Layers on Solids pogranichaye 8101 aa telakh vrashcheniya).
Gauss,
No. 2.
of
367-310.
1960.
(Teoriya trekh(‘Problemy
problems
239-296. Revolution
Symp.
Problcna
1959. (Podobaye
Three-dimensional
pogranichnogo
sloya
Gosenergoizdat, 5.
i
laninar
voprosy
248-256.
Timman. R. and Tsaat.
boundary
235
layers
MOSCOW -
teploperedachi.
Leningrad,
1960.
Y&A_. , klcthods
of
calculation
related
to
three-
(Xetody rascheta trekhmernogo laminarnogo pogranichnogo sloya). Symp. Problema pogranichnogo s loya i voprosy tep lopercdachi. Gosenergoizdat, MOSCOW - Leningrad, 360-371, 1960. dimensional
laminar
6.
Howarth. L.,
7.
Neiland,
8.
Physical
9.
Tables
Phi 1.
V.Ya.,
boundary
Msg.,
Inch.
42,
Zh.,
layers
No. 1,
2,
gas dynamics (Fizicheskaya Nauk SSSR, Moscow, 1959. of
thernodynanic
kikh funktsii
1433-1441,
31-36,
1951.
1961.
gazovaya
dinamika)
Izd-vo
Akad.
of air (Tablitsy termodinamicbesVTs Akad. Nauk SSSR, Moscow, 1962.
functions
vozdukha),
335.