differential models || the falkner-skan equation of boundary layer

6 The Falkner–Skan Equation of Boundary Layer

6.1 Introduction

The Falkner–Skan equation [4]

0)1( 2 =′−β+′′+′′′ ffff (6.1)

with boundary conditions

;0)0(;)0(;0 =′==η fff w

1)(; →∞′∞→η f ,

(6.2)

describes the class of so-called similar laminar flows in boundary layer on a per-meable wall (Fig. 6.1) and at varying main-stream velocity.

The dependent variable f is a dimensionless stream function, the independent

variable η is a dimensionless distance from the wall, a so-called similarity vari-

able. The first derivative of f with respect to η, i.e. f ', defines the dimensionless velocity component in x-direction, the second one, i.e. f '' defines the dimen-sionless shear stress in the boundary layer.

Solutions of the boundary problem defined by Eqs. (6.1), (6.2) create the theo-retical background for the analysis of friction resistance and heat or mass transfer for the following practical problems:

�

flows along curvilinear profiles, such as gas turbine blades or airplane wings,

�

flows on permeable (perforated) surfaces with blowing or suction, �

flows with condensation or evaporation on interfaces, �

flows with intensive catalytic reactions on a wall.

y

x

u∝∝∝∝

v(x,y=0)

δ (x)

Fig. 6.1. Velocity boundary layer on permeable wall

96 6 The Falkner–Skan Equation of Boundary Layer

The mass-transfer parameter fw in the boundary condition sets the measure for the mass flow rate through the wall boundary in either direction. Positive values determine flows with suction, negative with blowing (Fig. 6.1) through the wall boundary. The zero value corresponds to flow along impermeable wall with zero mass transfer.

The numerical parameter β (positive or negative) in the Falkner–Skan equation sets a degree of acceleration or deceleration of main stream. The flows with zero value for this parameter will be considered, i.e. the flows without longitudinal pressure gradient in main stream.

The Falkner–Skan equation (6.1), together with similar Eq. (6.22) for thermal boundary layer (see Sect. 6.7), constitute the theoretical basis of convective heat and mass transfer.

Both ordinary differential equations (6.1) and (6.22) mentioned above are ob-tained from more common partial differential equations of pulse and energy transport (1.32)–(1.35) using two fundamental ideas: boundary layer and similar-ity transformations.

1. Boundary layer. When low-viscous ( � / � � 0) fluids flow over body sur-faces, the cross changes of velocity should be concentrated within a boundary

layer � near the wall so that the viscous force might ensure the necessary decelera-

tion (loss of pulse) of main stream from u∞ down to zero at the wall, as can be shown by estimations of order in continuity and motion equations from Sect. 1.4:

� �∞

δ≈�=

δ

∂

∂+

∂

∂u

xv

v

y

v

x

u

x

u;0

(6.3)

�

ν∞

≈δ�

�

δ

∞ρµ

+∞≈∞∞

�

�µ

δ

∞µ

∂

∂µ+

∞µ

∂

∂µ+

∞ρ

∂∂

−=

δ∞δ∞ρ

∂∂

ρ+

∞∞ρ

∂∂

ρ

xux

u

x

u

x

uu

u

y

u

x

du

x

u

x

u

x

p

u

x

u

y

uv

x

uu

x

uu

1

2

2

2

2

2

2

2

2

2 ��

(6.4)

Here, we assume ( � / ) � 0, but not vanishing entirely, because the viscous force O( � u � / � ) should exist in our problem, to ensure adhesion to the wall. This is rel-ized by very thin boundary layer � � 0, � << x.

6.1 Introduction 97

An unacceptable alternative would be to generally neglect viscosity forces. Thus the second (i.e. the higher) derivatives would vanish. The mathematical de-gree of differential equation would be depressed, and the zero velocity condition on the wall could not be realized.

From (6.4) apparently follows that the indistinct expression about “low-viscosity” fluid may be replaced by the stricter statement about flows in the

asymptotical case of large Reynolds numbers Rex ≡ u∞x/ν >> 1. From (6.3), (6.4) the asymptotical simplified equations for velocity boundary layer follow:

;0=∂

∂+

∂

∂

y

v

x

u(6.5)

2

21

y

uv

dx

dp

y

uv

x

uu

∂

∂+

ρ−=

∂∂

+∂∂

. (6.6)

Within the limits of boundary layer δ/x << 1 (Fig. 6.1), the longitudinal velocity u

quickly varies from u∞ down to zero on the wall, and transversal velocity v re-

mains very small, v << u∞. The pressure variation over the cross-section can be neglected; only its longitudinal changes have to be taken into account.

The following differential equation for thermal boundary layer is derived from energy equation (1.35) in an analogous manner:

ca

y

ta

y

tv

x

tu

ρ

λ≡

∂

∂=

∂

∂+

∂

∂;

2

2

. (6.7)

2. Similarity transformation. Now, similarity transformation allows us to pro-ceed from partial differential equations to ordinary differential equations. Two in-dependent variables x and y are combined into a unique independent variable� = y/ � (x), where � (x) represents boundary layer thickness (Fig. 6.1).

Thus, the similar variable �

is the distance from the wall, measured on the scale of boundary layer thickness.

After entering a flow function � as new dependent variable, such that

xv

yu

∂

Ψ∂−=

∂

Ψ∂= ; ,

the continuity equation (6.5) is automatically satisfied. The meaning of the new variable becomes clear by looking at relationships:

�=Ψ==Ψ

y

dyuconstxdyud

0

,,by

according to which the flow function determines a fluid flow rate within the limits of the boundary layer.


The scale for Ψ may be u∞δ, where boundary layer thickness δ, as we saw earlier (Eqs. (6.4)), is evaluated as:

xx Re

1≈

δ.

So, the idea is to proceed from variables u(x, y), v (x, y) to similar variables ( � /(u� � ),

�). The efficiency of such replacement depends not only on differential

equations, but also on the structure of the boundary conditions. Generally, at the exterior boundary the distribution of main-stream velocity

u � (x) must be prescribed, as well as the cross mass flow rate at the wall boundary, also as function of x. In some cases (for example, at the power-behaved change of main-stream velocity, at the cross mass flux as reciprocal square root from x, at the power-behaved change of temperature drop in heat transfer problems), similar

(automodel) solutions of such boundary problems can be obtained. Let us briefly write the designations for similar flows:

,mxcu ⋅=∞

where

)/( 2∞ρ−≡ ux

dx

dPm

is modified Euler number.

Parameter β in Falkner–Skan equation is connected to Euler number by:

.1

2

+=β

m

m

The similar variable �

and the dimensionless stream function, f = � /(u � � ), are given as:

;1

dx

duy ∞⋅

νβ=η .

1

dx

du

uf ∞

∞

⋅νβ

Ψ=

Thus, due to asymptotic model of boundary layer and to similar transformations, the complicated original formulation (1.32)–(1.35) with partial differential equa-tions is reduced to the ordinary differential equation problem (6.1), (6.2). How-ever, even being the result of essential simplifications, the Falkner–Skan equation remains a complicated mathematical object, due to nonlinearity and boundary conditions mode [22, 54]. We will show now, that Mathcad provides the efficient tools to solve such complex problems.

In addition, our educational purposes are: solution of the applied problem and elaborate the necessary technique.

It is important to understand, how the velocity and temperature fields near the wall are established, how the flow resistance and heat transfer rate are to be calcu-lated and how the process can be controlled. We shall reach this goal by varying

6.2 Model Construction 99

control parameters, such as main-stream velocity, properties of fluid, rate of blow-ing or suction, and representing the results visually.

The principal technique issue will be the numerical solution of a two-point

boundary problem for the system of ordinary differential equations using the built-in Mathcad function sbval. The design infrastructure includes integration with methods rkfixed and odesolve, evaluation of the radical by root, interpolation bycspline, interp, matrix operations with stack, matrix, etc.

6.2 Model Construction

As velocity profile and shear stress, i.e. the first and second derivatives f, are of main interest, it is expedient to present differential equation (6.1) of the third order as a system of three first-order differential equations. The required function, its first and second derivatives are considered as components of vector-function (F0,

F1, F2). After apparent substitutions:

10 FFf =η∂∂

=η∂∂

; 212

2

FFf =η∂∂

=η∂

∂; 23

3

Ffη∂∂

=η∂

∂;

0Ff =

we receive the vectorial equations (6.8) instead of (6.1):

( )FDF ,η=η∂∂

, (6.8)

where the dependent variable and the right-hand side are defined as vector func-tions:

( )( ) ��

��

��

�

−β−−

=η

��

�

��

�=

2120

2

1

2

1

0

1

,;

FFF

F

F

FD

F

F

F

F . (6.9)

Because this system of differential equations is nonlinear (contains quadratic

terms F0⋅F2 and F12), numerical integration is required.

Boundary conditions (6.2) should be rewritten in new denotations as follows:

( ) ( ) ;00;0 10 == FfF w

( ) 11 =∞F .

(6.10)

The first of these conditions represents the mass flux through the wall bound-ary. The quantity fw is the given numerical parameter.

The second condition sets longitudinal component of velocity to zero at the

wall (at η=0) corresponding to the fundamental adhesion condition.


The third condition determines the longitudinal velocity on infinity as the main-stream velocity outside the boundary layer. At numerical integration, this condi-

tion is given on some large enough, but finite coordinate η, η=ηinf, such that its further increase does not result in any deformation of velocity profile. That can be achieved during several trial numerical experiments. One can also

formalize the selection ηinf so that the computational program itself would deter-

mine the necessary value of exterior boundary ηinf, satisfying the given calculation accuracy.

The purpose of further calculations is to obtain the flow field near the wall by different values of parameter fw : negative, appropriate to blowing (or to evapora-tion), and positive for suction (or condensation).

As to the second parameter β, we shall limit our further investigation to its zero value, i.e. with zero velocity-gradient in the main stream outside the boundary layer.

By numerical integration of system (6.8), the functions F0(η), F1(η), F2(η) will be obtained. Then the flow field in physical coordinates x,y should be calculated and graphically represented, and also shear stress at the wall, i.e. hydrodynamic resistance is evaluated.

Relationships between physical variables (i.e. longitudinal u(x,y) and transver-

sal v(x, y) components of velocity, flow function ψ(x,y)), on the one hand, and

similar Falkner–Skan variables (i.e. the dimensionless flow function f(η) and the

dimensionless distance from wall η) on the other, are given by following formu-las:

Longitudinal velocity:

1Ffu

uU =

η∂∂

==∞

. (6.11)

Transversal velocity:

X

FFff

Xu

vV

LL Re2Re2

1 01 −η=

��

��

�

−η∂∂

η==∞

. (6.12)

By zero value for η, eq. (6.12) determines connection between flow function at the wall and blowing or suction velocity:

( )X

Ff

Xu

v

L

ww

L Re2Re2

1 ,0−=−=

∞

. (6.13)

The second derivative calculated at the wall yields the friction coefficient:

X

F

X

f

u

y

u

u

c

L

w

L

wwwf

Re2Re22

,2''

22==

ρ

∂∂

µ

=ρ

τ=

∞∞

.(6.14)

6.3 Boundary-Value Problem. Method sbval 101

The self-similar variable �

L

X

Y

Re2

=η .(6.15)

is evaluated by means of physical coordinates (x, y). The dimensionless flow function f (or F0) is connected to the original flow

function ψ by:

XLuf L

2

Re

∞

Ψ= . (6.16)

Formulas (6.15), (6.16) are obtained for the case of constant main-stream ve-locity. The common relations for the similar variables ( � , f) have been given at the end of Sect. 6.1.

Let us remember the physical meaning of flow function:

�=ψ−ψ

2

112

y

ydyu , (6.17)

saying that the difference of values on two streamlines in a stream cross-section is equal to the volume flow (Fig. 6.5).

Reynolds number is determined by distance parallel to the wall L:

ν= ∞Lu

LRe .

Physical coordinates (x, y) are normalized by reference to length L:

X=x/L, Y=y/L.

6.3 Boundary-Value Problem. Method sbval

Boundary conditions (6.2) for the Falkner–Skan equation are given on both bor-

ders of integration interval: at the wall, η = 0, and far from it, at η → ∞, in fact,

however, in some large, but finite distance η = ηinf.

In other words, it is necessary to decide a boundary problem, or as it is some-times called, two-point boundary problem. Numerical algorithms for such prob-lems are more complicated than for an initial problem when all boundary conditions are given in an initial point.

For initial problems, the computational mathematics disposes of efficient nu-merical methods. The Runge–Kutta process with the fixed or adapted step is mostly used. It can be applied, reducing the boundary problem to the integration of series of initial problems with trial values of missing initial conditions in initial

point. Then, in end point, one can measure the “distance” between calculated


(trial) value of dependent variable and the one prescribed by boundary condition, and then judge the success or failure of each such attempt.

The analogy to shooting becomes obvious when the aim is adjusted by distance of the hit from the target. The numerical method, grounded on this analogy, is named “shooting method”.

Let us discuss this idea more concretely with reference to our problem. At ini-tial point, two conditions are given, i.e. values F0(0) and F1(0). Missing initial

condition is value F2(0) in initial point, instead of this the value F1(ηinf) = 1 at end

point is given. The initial problem can be solved with initial condition F2(0) = ξ,

where ξ is trial value taken by guess. As a result, some value F1,inf will be obtained at the end point.

It is unlikely that the correct value F1,inf =1 will be received after first integra-

tion already, but obtained value can be considered as function of trial initial ξ.Now it is clear how to proceed further.

There are two options:

• To solve equation F1,inf (ξ)−1=0, for example, by secant method, realizing

that trial values F1,inf (ξ) are not evaluated by simple substitution in some formula, but are obtained by numerical integration of a system of differ-ential equations

• To minimize the discrepancy (F1,inf (ξ)−1)2 → 0 using any optimization algorithm, for example, the coordinatewise optimization, or the simplex Nelder–Mead method, or any other available method.

In Mathcad there is a built-in function sbval that helps to find the missing con-

ditions at initial point. We do not really know the interior structure of this func-tion, but apparently, it works on one of the versions mentioned above. In Mathcad reference system the title sbval is decrypted as follows. The part “bval” corre-sponds to “boundary value”. Character “s” means “shooting method”.

As users, we can be content with this common view, but we should know how to write the call to this function (Fig. 6.2). As soon as the missing initial value will be returned by sbval function, the initial problem can finally be solved with the numerical method (for example, rkfixed).

The set of numerical parameters in our example will be as follows: �

Numerical parameter of problem giving acceleration of main stream: β=0 (for uniform main stream)

�

Numerical parameter of problem giving blowing (fw<0) or suction (fw >0): fw=0 (zero mass flux through the wall boundary)

�

The fixed zero value of coordinate at the wall (service parameter): ηw=0�

The numerical parameter giving exterior boundary(service parameter):

ηinf =6. Use of built-in function sbval is demonstrated in Fig. 6.2. The sequence of op-

erations is as follows: �

The parameter values are entered, the right-hand side (6.9) of equation set (6.8) is written

�

The vector of initial conditions according to (6.10) is created by user's function SetInit

6.4 The Solution of the Initial Problem. Method rkfixed 103

�

The trial value of the second derivative at η=0 is entered as component of

vector with zero index ξ0. In other problems, more than one initial condi-

tion may be missing, therefore ξ should be vector. �

User's function discrepancy evaluates the discrepancy at the end point of integration interval (F1–1). It should be equal to zero, see (6.10).

�

Function call sbval is written, and the output, i.e. the missing initial con-dition, is introduced in vector MissingInitCond. We shall remember, that this missing condition is the value of the second derivative at the wall, i.e. F2(0).

β 0:= ηw 0:= η inf 6:=

D η F,( )

F1

F2

F0− F2⋅ β 1 F1( )2−

�� ⋅−

��

� ��:=

fw 0:= ξ0 0.1:=

SetInit ηw ξ,( )fw

0

ξ 0

��

�:=

discrepancy ηw F,( ) F1 1−:=

MissingInitCond sbval ξ ηw, η inf, D, SetInit, discrepancy,( ):=

MissingInitCond 0.4696( )=

Fig. 6.2. Application of built-in function sbval returning the missing conditions at initial

point (F2(0) = 0.4696 in case of zero mass transfer)

6.4 The Solution of the Initial Problem. Method rkfixed

As all initial conditions are known now, it is possible to call any built-in function of numerical integration, for example rkfixed (Fig. 6.3). Parameters of this func-tion are: vector of initial conditions InitCond, coordinates of initial and end points, number of integration steps N and vector function of right-hand sides of system of differential equations. We shall remark, that the fragment of evaluations in Fig. 6.3 is the prolongation of the Mathcad document in Fig. 6.2, where the right-hand side D was already submitted.


InitCond SetInit ηw MissingInitCond,( ):=

InitCondT

0 0 0.4696( )=

N 200:= S rkfixed InitCond ηw, η inf, N, D,( ):=

η S0

� �:= f S

1� �

:= U S2

� �:= Stress S

3� �

:= �Uarray stack U

TU

T,( ):=

Zero i j,( ) 0:=

Vas_0 matrix rows Uarray( ) cols Uarray( ), Zero,( ):=

Uarray Vas_0,( )

fw 0=

Fig. 6.3. Longitudinal velocity distribution in the boundary layer on an impenetrable sur-

face

The output of integration is returned as array S, the column vectors of which are independent variable � , dimensionless flow function f, longitudinal component velocity U and shear stress, as shown in line 4. The velocity profile is constructed in such manner that the wall is identified by the horizontal coordinate, and the stream is considered moving from left to right. On vertical axis the dimensionless

distance � from the wall and on horizontal axis velocity U are represented.It deserves attention that the graph in Fig. 6.3 is constructed as Vector Field

Plot but so many horizontal arrows of velocity are drawn, that they merge creating the black area as distribution diagram. The last three lines above the diagram in Fig. 6.3 produce the data arrays required for diagram by built-in Mathcad-functions stack and matrix.

In Fig. 6.3, the classical velocity profile is shown in boundary layer on imper-meable surface without cross mass flow through the wall. In the following, the calculations for permeable surface will be carried out, and we shall see, how blow-ing or suction control the velocity profiles in boundary layers. But beforehand, we present the output in two different manners (Fig. 6.5). At first, we construct a vis-ual pattern of the vector flow field, and, secondly, we produce a diagram of the streamlines. It is useful to compare these pictures created by computer to photos of flows in album [57].

6.5 Flow Field Imaging 105

6.5 Flow Field Imaging

For visualization of flow field in original physical coordinates it is necessary to fill the flow area by a square net of coordinates (x, y), to calculate the values of the similar variable for each mesh point by formula (6.15), and then to find the values of longitudinal U and transversal V velocity projections, and also the flow func-tion � from Eqs. (6.11), (6.12), and (6.16)

�. These evaluations are realized in the

Mathcad-program by interpolation using built-in function linterp as logical centre (Fig. 6.4).

Fields S Re L, X min,( ) X max 1← nX 10← nY 40←( )

Ymax max S0

� �( ) 2

Re L⋅← Y min 0←

��

Xi X min iX max X min−

nX 1−⋅+←

Yj Ymin jYmax Ymin−

nY 1−⋅+←

η i j, YjRe L

2 Xi⋅←

Ui j, linterp S0

� �S

2� �

, η i j,,( )←

fi j, linterp S0

� �S

1� �

, η i j,,( )←

Vi j,1

2 Re L⋅ Xi⋅Ui j, η i j,⋅ fi j,−( )⋅←

Ψ i j, fi j,2 Xi⋅

Re L⋅←

j 0 nY 1−..∈for

i 0 nX 1−..∈for

U V Ψ Ymax( )

≡

Fig. 6.4. Interpolation routine to create flow field in physical coordinates

The results are shown in Fig. 6.5. On the leading edge of the wall (at X=0), the solution has a singularity, and construction of the graph starts with some small fi-nite value Xmin.

Two-dimensional arrays of U and V are defined on uniform grid, and the Vec-tor Field Plot diagram is applied to construction of the flow field (upper figure). Flow function is also calculated as two-dimensional array on the same grid. Constant values correspond to streamlines, and for their construction, diagram Contour Plot is applied (lower figure).


fw 0= ReL 100:= Xmin 0.01:=

U V Ψ Ymax( ) Fields S ReL, Xmin,( ):=

U V,( )

Xmin 0.01= Ymax 0.849=

Ψ

Fig. 6.5. Vector flow field and flow function for the impermeable wall

As is visible in the upper figure, the boundary layer becomes gradually thicker downstream. Streamlines are directed almost parallel to the wall, although some upward deviation is noticed due to deceleration, as fluid is “adhering” the wall (lower figure). Both longitudinal and transversal velocity components are zero on the solid impermeable surface.

The drag coefficient cf on the impermeable flat plate can be expressed by Eq. (6.14), in which the following substitution must be made using above obtained value of MissingInitCond (see Fig. 6.2) for the second derivative at the wall:

( ) 4696.002'' =≡ Ffw .

6.6 Boundary Layer on Permeable Walls 107

Thus we receive:

ν

=

ν

=∞∞ xuxu

c f 332.01

2

4696.0

2.

(6.18)

6.6 Boundary Layer on Permeable Walls

At intensive blowing through permeable walls (see Fig. 6.6, Fig. 6.7, and Fig. 6.8), an abrupt change of flow structure in the boundary layer is observed. The velocity profile (Fig. 6.6) turns to the characteristic S-shaped form. The shear layer appears to be pushed aside from the wall. The phenomenon of flow separation appears.

The shear stress at the wall decreases by actuation of blowing and will finally become zero at so-called critical blowing. The value of permeability parameter fw = –0.7 taken in this numerical example, is close to critical, and the friction coef-

ficient practically becomes zero (f″w = F2(0) = 0.05458 , see Fig. 6.6, vs.

f″w = F2(0) = 0.4696 for impermeable wall). Let us note, that the evaluation of the missing initial condition F2(0) by means of function sbval was carried out just as shown in Fig. 6.2 for the case of the impermeable wall, but with a modified set of parameters: ( � = 0, fw = – 0.7), instead of ( � = 0, fw = 0).

InitCond SetInit ηw MissingInitCond,( ):=

InitCondT

0.7− 0 0.054577( )=


η S0

� �:= f S

1� �

:= U S2

� �:= Stress S

3� �

:=

Uarray stack UT

UT,( ):= Zero i j,( ) 0:=


Uarray Vas_0,( )

fw 0.7−=

Fig. 6.6. Longitudinal velocity distribution in the boundary layer at blowing (fw = –0.7,

f″w = F2(0) = 0.05458)



U V,( )

Xmin 0.2= Ymax 0.849=

Ψ

Fig. 6.7. Vector flow field and flow function at blowing

U V,( )

Fig. 6.8. Vector field of velocity near the wall at blowing


Flow in boundary layer with suction is presented in Fig. 6.9, Fig. 6.10, and Fig. 6.11. We observe essential differences compared to the case of the impermeable wall. Now the boundary layer is pressed to the wall. The boundary layer thickness decreases noticeably in comparison with an impermeable surface and, especially, with flow under intensive blowing.

Since the boundary layer becomes very thin, the second derivative of flow function or, being the same, the first derivative of velocity becomes very large. Therefore, the shear stress will be increased by actuation of suction as it is visible from (6.14). The second derivative at the wall, proportional to shear stress, be-

comes f″w = F2(0) = 7.0692 (by permeability parameter fw = 7). It is a very large

value compared to f″w=F2(0) = 0.4696 for the original case of impermeable wall. Let us note, that the evaluation of the missing initial condition F2(0) by means

of function sbval was carried out just as shown in Fig. 6.2 for the case of imper-meable wall, but with a changed set of parameters: ( � = 0, fw = 7) instead of ( � = 0, fw = 0).

To see the flow structure near the wall in more detail, the vector velocity field is made scaled-up in Fig. 6.8 (blowing) and Fig. 6.11 (suction). As always, due to adhesion condition, the longitudinal component at the wall equals zero. The trans-versal velocity component, in contrast to the impermeable surface, is nonzero (positive for blowing and negative for suction). Therefore, the velocity vector at the wall is nonzero and is directed normal to the surface.

InitCond SetInit η w MissingInitCond,( ):=

InitCondT

7 0 7.069198( )=


η S0

� �:= f S

1� �

:= U S2

� �:= Stress S

3� �

:=

Uarray stack UT

UT

,( ):= Zero i j,( ) 0:=


Uarray Vas_0,( )

fw 7= �

Fig. 6.9. Longitudinal velocity distribution in the boundary layer at intensive suction

(fw = 7, f″w = F2(0) = 7.0692)


fw 7= ReL 100:= Xmin 0.2:=


U V,( )

Xmin 0.2= Ymax 0.849=

Ψ

Fig. 6.10. Vector flow field and flow function at suction

U V,( )

Fig. 6.11. Vector field of velocity near the wall at suction


In three numerical examples considered above (zero cross mass flux, blowing, suction), the parameter varied was the value of flow function fw on the wall, also called parameter of permeability (see Eq. (6.13)).

Each time, at the numerical integration of the boundary value problem for the

given parameter fw, the appropriate value of second derivative on the wall fw″ was obtained. The results are shown graphically in Fig. 6.12 in the form of functional

dependence fw″(fw).Apparently, we discover from the graph the asymptotic fw″ � fw for large posi-

tive parameter fw, i.e. for intensive suction.

1 0 1 2 3

1

1

2

3

f''w

fw

fw

f''w

fw

�� asympt

1

Fig. 6.12. Dependence of stress parameter fw" from mass flux parameter fw

Assuming this asymptotic in Eqs. (6.13) and (6.14) we obtain for intensive suc-tion:

∞

−=

u

vcwf

2(6.19)

or, identically:

[ ] ∞⋅−⋅ρ=τ uvww )( . (6.20)

The physical meaning of the asymptotic formula (6.20) can be explained as fol-lows: the part in square brackets is the cross mass flow from exterior main stream into the permeable wall, and the second factor is the longitudinal impulse, con-tained in each mass unit of cross flow. The product of both factors produces a momentum flux through the wall boundary which is equivalent to the stress at the wall.

Remarkable feature of the asymptotic law (6.19) or (6.20) is independence of stress (or friction coefficient) from viscosity of fluid (it is useful to compare the


asymptotic formula (6.19) with equation (6.18) for viscous friction on the imper-meable wall).

Another important result already considered above is zero stress value on the wall by intensive blowing (Fig. 6.6). This case of boundary layer separation is seen on the graph (Fig. 6.12) in the area of extreme negative values fw of blowing

where the wall velocity gradient fw″ will be zero.

6.7 Thermal Boundary Layer. Heat Transfer Law

If the temperature difference between wall and fluid (Fig. 6.1) is nonzero, there is a heat flux through the boundary layer that in engineering calculations is deter-mined by the Newton–Richman law:

( )∞−α= ttq ww , (6.21)

where α is the heat transfer coefficient, a measure of heat exchange intensity be-tween wall and fluid, an intricate function of velocity and flow regime, geometry, thermal properties of fluid. Finding this function is the main practical application of the theory of convective heat transfer.

Thermal Boundary Layer Equation

Let us remember, that the partial differential equations for velocity boundary layer (6.5), (6.6) after similar transformations turn into ordinary differential equa-tions (6.1), whose integration yields the full information on velocity and stress distribution.

Analogously, the partial differential equation (6.7) for thermal boundary layer is reduced to following ordinary differential equation:

0)()(Pr)( =η′⋅η⋅+η′′ gfg , (6.22)

with boundary conditions

1)(0)0( =∞= gg . (6.23)

The integration of this boundary problem yields the temperature and heat flux dis-tribution and eventually the heat transfer coefficient.

The similar variable � has been defined previously, see (6.15). The required function g represents the dimensionless temperature in the fluid flow:

w

w

tt

tyxtg

−

−=

∞

),(. (6.24)

Numerical parameter Pr (Prandtl number) is the ration of kinematic viscosity to thermal diffusivity of fluid.

6.7 Thermal Boundary Layer. Heat Transfer Law 113

Simple calculations show that heat transfer coefficient� � and Nusselt number Nu as its dimensionless form are expressed by following formulas which can be used after integration of the boundary problem (6.22), (6.23):

x

ug

yd

dg

tt

y

t

tt

qw

yw

y

w

w

νλ=

∂η∂

ηλ=

−

∂∂

λ−

=−

=α ∞

+=∞

+=

∞ 2

'

0

0;

ν=

λ

α= ∞xugx

Nu wx

2

'

.

(6.25)

For integration (6.22) we need a solution for the flow field. Only flow function f (but not its derivatives) is needed immediately, therefore it is convenient to use built-in function Odesolve for integration , as shown in Fig. 6.13. Let us restrict the problem to heat exchange on impermeable surface, therefore fw = f(0) = 0.

0 0.5 10

5

10

η

η inf

U η( )0 5

0

5

10

η

fa η( )

η inf 3.472=η inf root 0.99 U η( )− η, 1, 10,( ):=U η( )η

fa η( )d

d:=

fa η( ) fspl η( ) η 6≤if

η 6 fspl 6( )−( )− otherwise

:=

fspl η( ) interp Spl X, Y, η,( ):=Spl cspline X Y,( ):=

Yi f Xi( ):=Xi6 i⋅

N:=i 0 N..:=

f Odesolve η 6, N,( ):=

f' 6( ) 1f' 0( ) 0f 0( ) 0f''' η( ) f η( ) f'' η( )⋅+ 0Given

N 200:=

Fig. 6.13. Velocity boundary layer on impermeable wall (solution with Odesolve)


Referencing solver Odesolve takes only two lines of the program text. The computed distribution f is stored for further use in the form of spline-function fspl

obtained with the built-in functions cspline and interp.To present clearly the area of boundary layer, the velocity distribution is con-

structed and the distance from the wall � = 3.472 is marked on which 99% of all longitudinal velocity variation is completed. For finding of this boundary, the method root was used to solve the non-linear equation U( � )=0.99. Outside of layer � = 3.472, the velocity is practically constant already, and flow function var-ies linearly with distance from the wall. Good approximation for the flow function at any given � will be user's function fa, which is applied further at integration of the thermal boundary layer equation.

The integration of Eq. (6.22) and temperature distribution in the thermal boundary layer are demonstrated in Fig. 6.14. In the graph, the horizontal axis merges with the wall, and this axis also represents dimensionless temperature g.On the vertical axis, dimensionless distance � from the wall is plotted. So we have the temperature profile in common presentation.

Pr 1:=

Given

g'' η( ) Pr fa η( )⋅ g' η( )⋅+ 0 g 0( ) 0 g 6( ) 1

gPr_1 Odesolve η 6, N,( ):=

0 0.2 0.4 0.6 0.80

2

4

6

8

10

Pr=0.1Pr=1Pr=10

Wall

Temperature

Dis

tan

ce

Fig. 6.14. Temperature distribution for various Prandtl numbers


The calculations are carried out for a large interval of Prandtl numbers. It should be remembered that values of the Prandtl number close to unit are charac-teristic for heat transfer with gases, much larger values for viscous low-conductivity organic liquids, much smaller unit values for molten metals. The graph in Fig. 6.14 shows, that the separation of heat-carriers in three such groups has a clear physical meaning. To distinguish between the results for different val-ues of the Prandtl number, we have supplied g with an appropriate literal sub-script.

• In case Pr = 1, the profiles of velocity and temperature completely coin-cide and the thicknesses of hydrodynamic and thermal boundary layers are identical. Therefore, the curve labeled as Pr = 1 should also be con-sidered as velocity profile for all of the three cases shown.

• At Pr << 1 (see numerical example with Pr = 0.1), the thermal boundary layer thickness exceeds by far that of velocity. Therefore, velocity will be constant within most part of the thermal boundary layer, except in the very close neighborhood of the wall.

• At Pr >> 1 (see numerical example with Pr = 10), quite the contrary oc-curs, and the thermal boundary layer only exists at the bottom of the ve-locity boundary layer. As a consequence, velocity will be low throughout the thermal boundary layer, diminished by proximity to the wall.

The results of calculations at different Prandtl numbers are collected in the ta-ble of Fig. 6.15, where the first line indicates numbers 2 to 11 of the 10 variants, the second gives the Prandtl numbers are indicated, and the third the pertaining values of the dimensionless temperature gradient g � w.

These results are also presented as graphs containing the following data:

• results of numerical integration (circles),

• asymptotic solutions for large (dashed line) or small Prandtl numbers (dotted), respectively

• approximated general-purpose equation g'appr for wall temperature gradi-ent g'w (solid line, obtained by superposition of asymptotes, applicable in all ranges of Prandtl numbers).

Values g'w are substituted in formulas (6.25) to calculate a required value of heat transfer coefficient as a measure of heat exchange intensity.

Using this theory, we are able to calculate the heat transfer coefficient for three different heat-transfer mediums, mercury, air and oil, presenting three characteris-tic groups and also for the very important case of water (Fig. 6.16, fluids identified by indexes). Thermal properties are taken for standard conditions. Dimensional data are written in SI (kg, m, s, K): thermal conductivity � , W/(m K), kinematic viscosity � , m2/s; wall length L, m; main-stream velocity U, m/s; heat transfer co-efficient � , W/(m2 K).

The wall length and main-stream velocity were selected small as to guarantee laminar conditions.


Pr_gT

2 3 4 5 6 7 8 9 10 11

0

1

0.01 0.1 0.5 1 2 4 8 10 20 100

0.073 0.198 0.367 0.47 0.597 0.756 0.955 1.03 1.299 2.226

=

Pr Pr_g0

� �:= g'w Pr_g

1� �

:=

g'appr Pr( ) 0.479 Pr

1

3⋅

�� 4−

0.798 Pr

1

2⋅

�� 4−

+

� � �� 0.25−

:=

1 .103

0.01 0.1 1 10 1000.01

0.1

1

10

CalculPr >> 1Pr << 1approx

g'w

0.479Pr

1

3⋅

0.798Pr

1

2⋅

g'appr Pr( )

Pr

Fig. 6.15. Dimensionless temperature gradient as function of Prandtl number

The heat transfer coefficient is estimated by formula (6.25):

L

ugw

νλ=α ∞

2

'

.

The graphs in Fig. 6.16 demonstrate, how strongly the intensity of heat ex-change depends on physical properties of fluid. The heat transfer coefficient to mercury (molten metal with high thermal conductivity) on three orders exceeds the one to air. Evidently, water is a much better cooling agent, than viscous oil or air with its small density and thermal conductivity. Therefore, projects of replacing air-cooling of personal computers and even notebooks are seriously considered. It would make the cooling systems also silent.

Another important result of the theory is essential increase of heat exchange in-tensity with increasing main-stream velocity: the heat transfer coefficient is pro-portional to the square root of velocity (see (6.25)).


Hg

Air

Water

Oil

��

0

1

2

3

�� := λ

7.8

2.6 102−

⋅

0.6

0.15

��

� ��:= ν

1.2 107−

⋅

1.5 105−

⋅

1.0 106−

⋅

1.5 103−⋅

��

� ��:= Pr

0.03

0.7

7.0

1.6 104

⋅

��

� ��:=

g'appr Pr( ) 0.479 Pr

1

3⋅

�� 4−

0.798 Pr

1

2⋅

�� 4−

+

�� 0.25−

:=

α fluid U, L,( ) λ fluid

g'appr Prfluid( )2

⋅U

ν fluid L⋅⋅:=

L 0.1≡ U 0.1 0.2, 1..≡

0 0.5 11

10

100

1 .103

1 .104

α Hg U, L,( )

α Air U, L,( )

α Water U, L,( )

α Oil U, L,( )

U

Fig. 6.16. Convective heat transfer to different heat-transfer mediums

Though these results are obtained for laminar conditions, the qualitative content of deduced relationships also holds for turbulent flows that are more important in practice (although more complicated for calculation).

Heat Transfer Law

Eq.(6.25) will now be presented in greater generality than the one deduced for lon-gitudinal flow of the isothermal plate.

Let us rewrite (6.25) as a relationship linking

• the local value of heat transfer coefficient, made dimensionless by means of scaling with � cu ,

• and the Reynolds number constructed on local values of boundary layer thickness and main-stream velocity:


txx

reltxxwg

�u Re

1

Pr

,' δ⋅

=ρ

α

∞

, (6.26)

where

ν

δ= ∞ txx

txx

uRe ;

∞

ν

δ=δ

u

x

txxreltxx

2, .

Enthalpy thickness � txx may be calculated under formula (5.1), if profile of veloc-ity and profile of temperature are known. In dimensionless form:

( ) ( )( ) ηη−η=δ �∞

dgUreltxx 1

0

, .

Results of relative enthalpy thickness computed by this formula are presented as graphs in Fig. 6.17.

δxx_rel 0.4696:=

0 5 100

0.2

0.4δ txx_rel

i

δxx_rel

Pri

Fig. 6.17. Dimensionless enthalpy thickness as function of Prandtl number

For comparison, the value of the relative momentum thickness is marked in the same graph:

∞

ν

δ=δ

u

x

xxrelxx

2_ .

The thickness ratio of thermal and velocity boundary layers was considered above (see Fig. 6.14), and all numerical data needed for formula (6.26) were obtained al-ready (see Fig. 6.13 and Fig. 6.14). The result will be as follows:

6.8 Troubles with Odesolve 119

3/4Pr

22.0(Pr),

Re

(Pr)≅ϕ

ϕ=

ρ

α≡

∞ txxcuSt . (6.27)

Approximation function � (Pr) in (6.27) is suitable for gases and nonmetallic liq-uids.

Within the integral method of boundary layer (see previous chapter), the rela-tion (6.27) for Stanton number St is also called the heat transfer law at laminar condition. Let us accentuate that the formulation was obtained immediately for the limited conditions of heat exchange for the isothermal flat plates in parallel lami-nar flows.

Calculations and experiments show, however, that this law can also be applied with satisfactory accuracy for convective heat transfer on not isothermal surfaces and with variable main-stream velocity. A practical example of such problems was considered in the previous chapter.

6.8 Troubles with Odesolve

The built-in integrator, which appeared in Mathcad 2000 version, is intended for integration of ordinary differential equations, including two-point boundary problems. The Odesolve call (see e.g. Fig. 6.14) practically does not differ from the routine mathematical notation and looks much easier than the use of the sbvalmethod. Solution of the Falkner–Skan equation is written in one line. By reference to Odesolve, a function will be returned, but not an array, when using other solvers such as rkfixed etc.

The problem is, that manipulating with this return function the peculiar results may be obtained as is demonstrated in the examples of Fig. 6.18, Fig. 6.19.

The first error is identified by evaluation of the first and second derivatives at an initial point (Fig. 6.18). It is apparent, that the zero value of the first derivative given by the initial condition is not reproduced. For the second derivative, zero is obtained instead of the correct value (0.4696, see Fig. 6.3). It looks like a serious mistake, rather than an allowable error of solution.

Apparently, the return function f is not a routine function since it is not possible to apply standard mathematical operation of a series expansion.

Fig. 6.19 illustrates the peculiarities detected at attempts to sequentially apply the spline interpolation and then the derivation. The first fragment shows that at small interpolation steps (0.01) the same improper results are returned as by direct evaluation of the derivatives.

The second fragment shows, that interpolation steps may be selected (0.1) that yield satisfactory results but looking at the last fragment, countenance gets lost again, because minor step variation (0.09) gives rise to an abrupt change in the re-sult. The user could eventually find reasons why Odesolve behaves in this man-ner (it would be better, however, the producer to explain its features more clearly).


TOL 0=

Given f''' η( ) f η( ) f'' η( )⋅+ 0 f 0( ) 0 f' 0( ) 0 f' 6( ) 1

f Odesolve η 6,( ):= f function=

η 0:=η

f η( )d

d0.0135565=

2η

f η( )d

d

20= w

e

f η( ) series η 0, 2, →

Fig. 6.18. Odesolve and series expansion

2η

fs η( )d

d

20.417474=

ηfs η( )d

d2.225674 10

3−×=η 0:=

fs η( ) interp Spl X, Y, η,( ):=Spl cspline X Y,( ):=

Yi f Xi( ):=Xi 0.09 i⋅:=i 0 5..:=

__________________________________________________________

2η

fs η( )d

d

20.469747=

ηfs η( )d

d3.976486− 10

6−×=η 0:=


Yi f Xi( ):=Xi 0.1 i⋅:=i 0 5..:=

__________________________________________________________

2η

fs η( )d

d

21.781474 10

15−×=η

fs η( )d

d0.013557=η 0:=


Yi f Xi( ):=Xi 0.01 i⋅:=i 0 5..:=

Fig. 6.19. Odesolve and spline interpolation

6.9 Conclusion

Use of the blowing technique can protect the surfaces from action of high-temperature gas flows or from chemically hostile environments. Patterns of vector flow fields show, that by blowing through permeable walls under large negative

values of parameter fw = – 0.7 the phenomenon of separation of boundary layer takes place (Fig. 6.7, Fig. 6.8). In immediate proximity to the wall, the velocity is directed vertically, and not before some distance from the wall a longitudinal shear flow will be formed.

6.9 Conclusion 121

By varying the parameter value fw, the influence of mass flow through the boundary layer can be observed. At fw = 0 the standard flow pattern on imperme-able walls is received, and at positive values – the flow pattern with suction. In aerodynamics, the suction technique is used for prevention of the dangerous phe-nomenon of boundary layer separation by flow over a wing with large angle of at-tack, as is the case during the landing of aircrafts. In heat exchangers, problems with suction arise in case of intensive condensation of vapours.

Interesting effects may occur as happen combined effect of parameters of ac-

celeration β and permeability fw. Varying these two parameters, the engineer-designer can realize a boundary layer control, for example, applying the suction to prevent a flow separation in the case of an adverse pressure gradient (i.e. with flow in the direction of increasing pressure).

Solution of the Falkner–Skan equation and the similar equation for thermal boundary layer explores the basic function relations for hydrodynamic resistance and heat transfer, showing the role of flow rate, geometry, and the physical prop-erties of a fluid. Presented in the form of laws of resistance, heat and mass transfer for laminar flow (like Eq. (6.27)), the solutions are used within the integral method oriented on engineering applications [18, 44]. An example of such prob-lems has been considered in Chap. 5. Application for the boundary layer concept near the phase boundary is described in [40].

The main purpose of the present chapter in the sphere of computing technique is the solution of two-point boundary problems for systems of ordinary differential equations. We have achieved this by using the built-in function sbval based on the shooting method. But this method does not always work [22]. There are problems with a strong dependence of the solution on the initial data. Then numerical insta-bility will hinder convergence of the sbval method. Apparently, the built-in func-tion bvalfit as a special modification of the shooting method will be more stable. An appropriate example is considered in Sect. 10.6.

Another, more radical alternative would be the finite difference method in com-bination with TDMA (see Chap. 11). For systems of high order, the TDMA for tridiagonal block matrix may be used [1,22].

differential models || the falkner-skan equation of boundary layer

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