Durst F. Fluid Mechanics: An Introduction to the Theory of Fluid Flows

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472 16 Flows of Large Reynolds Numbers

This ansatz takes into consideration that η ≈

with δ ≈

√

νt =

νx

∞

can

be set. For the stream function one can write

Ψ =

dy = U

∞

νx

∞

F (η)dη =



∞

νxf(η). (16.44)

For the diﬀerent terms in (16.38) and (16.39), the following relationships can

thus be derived:

∂Ψ

∂y

dΨ

dη

∂η

∂y

= U

∞

dη

= U

∞



(η), (16.45a)

= −

∂Ψ

∂x

= −

∞

f(η)+

∂Ψ

∂η

∂x

= −

νU

∞

f(η) −



∞

νx



−





(η) (16.45b)

or rewritten for U

νU

∞

(ηf



(η) − f (η)) . (16.46)

For the further terms in (16.38) and (16.39), we can deduce

∂U

∂x

∂

∂x∂y

= −

∞



(η), (16.47)

∂U

∂y

∂

∂y

= U

∞

νx



(η) , (16.48)

∂

∂y

∂

∂y

∞

νx



(η) . (16.49)

On introducing all the derived terms into (16.42), one obtains

−

∞

ηf





∞

[ηf



− f ] f



= ν

∞

xν



(16.50)

or, after complete rearrangement:



+2f



=0. (16.51)

This is the ordinary diﬀerential equation derived by Blasius in his G¨ottin-

gen doctoral thesis. As he showed, it can be integrated numerically. This

integration results, with the following boundary conditions for f and f



η =0: f =0; f



=0 and η →∞: f



→ 1 (16.52)

16.3 Flat Plate Boundary Layer (Blasius Solution) 473

Fig. 16.4 The numerical solution of the

Blasius boundary-layer equation yields the

above-stated distributions of f(η), f



(η),



(η)

Fig. 16.5 Longitudinal and transversal velocity distributions in a boundary layer

in the distributions shown in Fig. 16.4 for f,f



and f



. The given functional

values can be employed at each point η, in order to compute U

∞

= f



(η)

and U

∞

=(1/2U

∞

)



νU

∞

/x(ηf



− f). The distribution of U

∞

presented in Fig. 16.5a and

∞



∞

in Fig. 16.5b. Both ﬁgures indicate

velocity distributions as they are also found in experimental investigations.

This is shown in Fig. 16.6. The above considerations have shown that, by

introducing the boundary-layer equations, it has been possible to handle an

important ﬂow theoretically, namely the viscous ﬂow over a ﬂat plate.

It is interesting to see from Fig. 16.4 that the U

velocity component at

the outer edge of the boundary layer, i.e. for η →∞, adopts the value

)

∞

=0.8604U

∞

. (16.53)

This velocity component directed out of the boundary layer ﬂow region comes

from the fact that with increasing length along the plate of the ﬂow and

thus increasing boundary layer thickness, the ﬂuid is being forced away from

the wall.

Further values concerning the velocity proﬁles of the ﬂat plate bound-

ary layer can be taken from Table 16.1, in which f(η),f



(η)andf



(η)are

474 16 Flows of Large Reynolds Numbers

Theoretical values of Blasius solution

Experimental results measured by LDA

Fig. 16.6 Agreement of experimental and theoretical results for a ﬂat plate boundary

layer

indicated. This table can be employed for the determination of all proper-

ties of the ﬂat plate boundary-layer ﬂow. The computed values are in good

agreement with the experiments, as can be seen from Fig. 16.6.

16.4 Integral Properties of Wall Boundary Layers

In the preceding considerations of boundary-layer properties, δ wasusedasa

quantity, without being deﬁned precisely. It was introduced into the consid-

erations from derivations of the molecular momentum diﬀusion as δ ≈

√

νt

with t

= t

=(x/U

∞

), so that the following holds:

δ ≈

νx

∞

. (16.54)

It is possible to make a somewhat more precise statement, of what δ is, by

means of the deﬁnition of the displacement thickness δ

, which indicates the

extent to which the ﬂow, originally arriving with U

∞

, was displaced by the

plate, due to the boundary-layer development (integral theorem for mean

properties):

∞

− U

)dy (16.55)

or rewritten in terms of δ

and integrated:

∞



1 −

∞



dy =

νx

∞

[1 − f



(η)] dη =1.73

νx

∞

. (16.56)

16.4 Integral Properties of Wall Boundary Layers 475

Table 16.1 Solution values of the Blasius equation according to Howarth

η = y

∞

νx



∞



0000.33206

0.2 0.00664 0.06641 0.33199

0.4 0.02656 0.13277 0.33147

0.6 0.05974 0.19894 0.33008

0.8 0.10611 0.26471 0.32739

1.0 0.16557 0.32979 0.32301

1.2 0.23795 0.39378 0.31659

1.4 0.32298 0.45627 0.30787

1.6 0.42032 0.51676 0.29667

1.8 0.52952 0.57477 0.28293

2.0 0.65003 0.62977 0.26675

2.2 0.78120 0.68132 0.24835

2.4 0.92230 0.72899 0.22809

2.6 1.07252 0.77246 0.20646

2.8 1.23099 0.81152 0.18401

3.0 1.39682 0.84605 0.16136

3.2 1.56911 0.87609 0.13913

3.4 1.74696 0.90177 0.11788

3.6 1.92954 0.92333 0.09809

3.8 2.11605 0.94112 0.08013

4.0 2.30576 0.95552 0.06424

4.2 2.49806 0.96696 0.05052

4.4 2.69238 0.97587 0.03897

4.6 2.88826 0.98269 0.02948

4.8 3.08534 0.98779 0.02187

5.0 3.28329 0.99155 0.01591

5.2 3.48189 0.99425 0.01134

5.4 3.68094 0.99616 0.00793

5.6 3.88031 0.99748 0.00543

5.8 4.07990 0.99838 0.00365

6.0 4.27964 0.99898 0.00240

6.2 4.47948 0.99937 0.00155

6.4 4.67938 0.99961 0.00098

6.6 4.87931 0.99977 0.

00061

6.8 5.07928 0.99987 0.00037

7.0 5.27926 0.99992 0.00022

7.2 5.47925 0.99996 0.00013

7.4 5.67924 0.99998 0.00007

7.6 5.87924 0.99999 0.00004

7.8 6.07923 1.00000 0.00002

8.0 6.27923 1.00000 0.00001

8.2 6.47923 1.00000 0.00001

8.4 6.67923 1.00000 0.00000

8.6 6.87923 1.00000 0.00000

8.8 7.07923 1.00000 0.00000

476 16 Flows of Large Reynolds Numbers

On now choosing δ =3δ

as a more precise deﬁnition of the thickness of the

boundary layer, as proposed by Prandtl, one obtains

δ =5.2

νx

∞

. (16.57)

According to Table 16.1, u(δ) shows a deviation of δ from the external velocity

∞

by about 0.5%.

Analogous to the above-computed displacement thickness, which was de-

ﬁned and computed as “mass loss thickness of the boundary layer”, the

momentum-loss thickness can also be deﬁned and computed. Here, the

introduction of the momentum deﬁcit into the integral took place with

∆U =(U

∞

− U

ρU

∞

= ρ

∞

− U

)dy. (16.58)

Solved in terms of δ

and integrated, this yields

∞



1 −

∞



dy =

νx

∞



[1 − f



]dy =0.664

νx

∞

. (16.59)

Hence, a comparison of the various thickness gives δ

=0.384,δ

=0.128δ.

It is important to take into consideration that the boundary-layer equa-

tions become valid only with a certain distance x from the front edge of the

plate, i.e. the boundary-layer equations hold only from a certain Re

, with

= U

∞

x/ν.

For smaller values of Re

, the complete Navier–Stokes equations have to

be employed to compute the velocity ﬁeld. The solutions of these equations

have to consider, moreover, that the front edge of the plate represents a

singularity, which requires special attention when carrying out numerical so-

lutions, as shown by Carrier and Lin [16.3], Boley and Friedman [16.2], Shi

et al. [16.7], etc.

A further quantity, which can be derived from the solutions of the Blasius

boundary layer equation, is the friction coeﬃcient c

, deﬁned as

|τ

∞

where |τ

| = local momentum loss to the wall and

ρU

∞

= stagnation pres-

sure of the outer ﬂow.

The local momentum loss |τ

| is computed as

|τ

| = µ



∂U

∂y



y=0

µU

∞



(0)



νx/U

∞

0.332

√

ρU

∞

(16.60)

16.4 Integral Properties of Wall Boundary Layers 477

and for c

for boundary-layer ﬂows it therefore holds that

0.664

√

. (16.61)

Often it is suﬃcient for boundary-layer ﬂows to state their integral

properties as indicated in (16.56), (16.57) and (16.59), i.e. the quantities

δ =5.2

νx

∞

boundary-layer thickness,

=1.73

νx

∞

displacement thickness,

=0.664

νx

∞

momentum-loss thickness

serve for indicating integral properties of boundary-layer ﬂows (Fig. 16.7).

General considerations of the integral form of the boundary-layer equa-

tions, that are based on (16.38) and (16.39), originate from von Karman. He

proposed to integrate the equation

∂U

∂t

+ U

∂U

∂x

+ U

∂U

∂y

= −

+ ν

∂

∂y

(16.62)

from y =0toy = δ(x), where (16.62), with the aid of the continuity equation,

can be rewritten as follows:

∂U

∂t

∂(U

)

∂x

∂(U

)

∂y

= −

∂

∂y

. (16.63)

Now applying integration from 0 to δ,oneobtains

Fig. 16.7 Illustration of the dis-

placement thickness δ

,themo-

mentum loss thickness δ

,andthe

boundary-layer thickness δ

478 16 Flows of Large Reynolds Numbers

∂

∂t

dy +

∂(U

)

∂x

dy +[U

]

= −

δ +

∂

∂y

dy. (16.64)

For

∂(U

)

∂x

dy we can write:

∂(U

)

∂x

dy =

dy −U

∞

dδ

. (16.65)

Moreover, introducing [U

]

= −U

∞

∂U

∂x

dy and rewriting:

∞

∂U

∂x

dy = U

∞

dy −U

∞

dδ

(16.66)

yields with

∂

∂y

dy = −

(x)

the following equation:

∂

∂t

dy +

dy − U

∞

dy = −

δ −

. (16.67)

For stationary ﬂows, with consideration of −

= U

∞

,onecan

therefore write:

dy −U

∞

dy − U

∞

= −

(16.68)

or, in a somewhat rewritten way:

dy−

∞

dy+

∞

dy−

∞

dy = −

(16.69)

so that the following equation can be deduced:

∞

− U

)dy +

∞

− U

)dy =+

. (16.70)

16.4 Integral Properties of Wall Boundary Layers 479

As outside δ, i.e. for δ →∞, there is no more change of the velocity proﬁle,

one can write

∞

− U

)dy



 

∞

− U

)dy



 

∞

. (16.71)

Therefore, the following integral relationship holds:

∞

. (16.72)

This equation can be rewritten in the following form:

dδ

+(2δ

+ δ

)

∞

ρU

∞

. (16.73)

For the Blasius boundary layer, this equation reduces to

dδ

. (16.74)

This result can be veriﬁed by inserting (16.59) and (16.61) into (16.74).

The basic idea behind the Karman integral consideration of the boundary-

layer equations is the fact that, for determining integral properties of

boundary-layer ﬂows, one does not require the exact distribution of U

∞



(η). When giving an approximate function U

∞

= g(y/δ), the general

character of boundary-layer ﬂows is already captured. The Karman integral

equations allow one to determine good approximations for δ(x)andc

(x):

∞

= A + Bη + Cη

+ Dη

(16.75)

with η = y/δ and the boundary conditions

y =0:U

=0 and

∂

∂y

=0,

y = δ : U

= U

∞

and

∂U

∂y

=0.

(16.76)

With these values, f



(η)=

η −

can be stated as the velocity proﬁle and

the values deduced:

δ =4.641

νx

∞

; c

1.293

√

; δ

=1.74

νx

∞

. (16.77)

These are close to the values derived from the Blasius solution of ﬂat-plate

boundary-layer ﬂow.

480 16 Flows of Large Reynolds Numbers

16.5 The Laminar, Plane, Two-Dimensional

Free Shear Layer

On allowing two parallel ﬂows of identical ﬂuids, that diﬀer only in having

diﬀerent ﬂuid velocities, to interact with one another, a ﬂow results that is

deﬁned as a laminar, plane, two-dimensional free shear layer. Such a ﬂow is

sketched in Fig. 16.8, which shows that the features of the ﬂow are generated

by the cross-ﬂow molecular momentum transport that occurs along the ﬂow.

The velocity gradient in the shear layer decreases with increasing distance

to x = 0. This takes place because of the momentum transport, i.e. because

of the momentum transport from the region of high velocity to the region of

low velocity, as indicated in Fig. 16.8.

The ﬂow sketched in Fig. 16.8 has properties which were employed for the

derivations of the boundary-layer equations for ﬂows.

• In the ﬂow direction, there exists a convection-dominated momentum

transport. In the cross-ﬂow direction, a diﬀusion-dominated momentum

transport occurs.

Likewise, the ﬂow has no pressure gradient, i.e.

∂P

∂x

∂P

∂y

= 0, so that one

can state

• The form of the boundary-layer equations, which was deduced by Bla-

sius for the plate boundary-layer, also holds for the laminar, plane,

two-dimensional free shear layer.

Thus, (16.42) has to be employed to treat free shear ﬂows of the kind

sketched in Fig. 16.8, i.e. one obtains the solution by the following diﬀerential

equation:

∂Ψ

∂y

∂

∂x∂y

−

∂Ψ

∂x

∂

∂y

= ν

∂

∂y

. (16.78)

( U )

Region of shear flows

Fig. 16.8 Formation of a laminar, free shear layer by momentum diﬀusion

16.6 The Plane, Two-Dimensional, Laminar Free Jet 481

Fig. 16.9 Diagram of the considered plane, two-dimensional, laminar free shear layer

With the ansatzes η = y



)

/νx and Ψ =



ν (U

)

xf(η), the

following ordinary diﬀerential equation results:



+2f



= 0 (16.79)

which has to be solved for the following boundary conditions, describing the

considered free shear-layer ﬂow:

η =+∞ : f



=1,

η = −∞ : f



= λ =

)

η =0: f =0.

(16.80)

The solution of (16.79) has again to be carried out numerically, similarly

to the Blasius solution for the ﬂat plate, as there is no analytical solution of

(16.79) available. The solution obtained by Lock [16.6] is indicated in Fig. 16.9

for λ = 0, i.e. (U

∞

)

=0,andalsoforλ =0.5.

16.6 The Plane, Two-Dimensional, Laminar Free Jet

Another ﬂow with boundary-layer character will now be investigated. This

ﬂow is usually referred to as a two-dimensional, laminar free jet, sketched in

Fig. 16.10.