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University of Groningen

Viscous-inviscid interaction with the quasi-simultaneous method for 2D and 3D aerodynamicflowCoenen, Edith Gerda Maria

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Publication date:2001

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Citation for published version (APA):Coenen, E. G. M. (2001). Viscous-inviscid interaction with the quasi-simultaneous method for 2D and 3Daerodynamic flow. [S.n.].

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Chapter 3

External ow region

3.1 Introduction

For the modelling of the ow outside the boundary layer an inviscid ow solver is needed.

As it is of primary importance in the work described in this thesis to develop a fast and

eÆcient computational method, suitable for a design-optimisation environment, a surface

singularity method for irrotational subsonic ow is preferred to a physical more complete

Euler or full-potential- ow solver.

In the next sections the potential- ow problem is discussed for which solutions can

be obtained by distributing singularity elements on the boundaries. First, the purely

inviscid potential- ow problem with no boundary layer present is examined. Later on,

the boundary-layer displacement e�ect is taken into account, by modelling an extra source

singularity distribution over the pro�le's surface and the wake. This is termed the viscous

formulation of the potential- ow problem.

The unknown strengths of the singularities are to be determined numerically. This

is done straightforwardly with a panel method, which requires the discretisation of the

geometry into a number of panel segments.

Various problems are discussed, of which the model problem of the ow over at plates,

which corresponds with non-lifting thin-aerofoil/wing theory, is described in greater detail.

The constant potential Dirichlet method is applied for the more practical problems of

modelling wing and aerofoil ow. For each of these problems the edge velocity equations

are determined, required for the coupled potential- ow/boundary-layer calculations.

3.2 Inviscid steady potential- ow problem

Consider an arbitrary body within an enclosed three-dimensional volume V , with the

contours of the body surface and wake, Sb and Sw, respectively, and the outer control

surface S1, surrounding the body and the wake surface, as shown in �gure 3.1.

The unit normal n is the vector pointing towards the inside of volume V and towards

the outside of Sb and Sw. The body, which has a body-�xed Cartesian coordinate system

(X; Y; Z), is subject to the in ow velocity Q1 = (U1; V1;W1)T .

The inviscid ow inside domain V over the solid body when assumed incompressible

30 Chapter 3 External ow region

∞QSb

Sw

(ξ,η,ζ)q

n

S∞

V, φ

n

Z

X

Y

p(X,Y,Z)

r

n

Figure 3.1: Nomenclature potential- ow problem.

and irrotational, can be treated as potential ow [71]. This is governed by the Laplace

equation

r2� = 0; (3.1)

for the velocity potential �(X; Y; Z).

On the solid body surface Sb in the ow regime there is zero normal velocity which

can be expressed as the Neumann boundary condition

r� � n = 0: (3.2)

On the outer control surface S1 the disturbance due to the presence of the body must

vanish in the limit and the potential must be equal to the undisturbed freestream potential

�1

limS1!1

(r��r�1) = 0: (3.3)

Once � has been obtained, the pressure follows from Bernoulli's equation.

3.2.1 Formulation of the integral equation

Using Green's identity the above constructed boundary-value problem can be transformed

into an integral equation for the potential � [50]

4�E(p)�(p) =

ZZSb+Sw+S1

��r

1

r�

1

rr�� n dS; (3.4)

with r(p; q) the distance between the point of integration q(�; �; �) and a �xed �eld point

p(X; Y; Z) at which the potential is calculated

r(p; q) =p(X � �)2 + (Y � �)2 + (Z � �)2: (3.5)

3.2 Inviscid steady potential- ow problem 31

The function E(p) in (3.4) is given by

E(p) =

8>>>><>>>>:

0; for p outside the ow �eld V ;

12; for p on the boundary of V ;

1; for p inside the ow �eld V :

(3.6)

If the ow �eld inside the body is considered, an equation similar to (3.4) can be derived

for the internal potential �in [50]

4�(1� E(p))�in(p) =

ZZSb+Sw

�1

rr�in � �inr

1

r

�� n dS: (3.7)

Equations (3.4) and (3.7) can be combined to a form relating � and �in

4�(E(p)�(p) + (1� E(p))�in(p)) = (3.8)

ZZSb+Sw

�(�� in)r

1

r�

1

rr(�� in)

�� n dS +

ZZS1

��r

1

r�

1

rr�� n dS:

The wake surface Sw is assumed to have zero thickness, such that the normal velocity

jump across the wake dividing streamline Sw is zero, while a jump in the potential is

allowed. With this condition the problem reduces to determining the values of � � �in

and r(� � �in) � n on the boundaries. The di�erence in external and internal potential

can be provided by a continuous doublet distribution of strength � over the surface of the

body and the wake. The di�erence between the normal derivatives of the external and

internal potential can be induced by a continuous source distribution of strength � on the

body

� = �� in; � = r(�� in) � n: (3.9)

Using the above de�nitions for � and �, the potential for points p inside the ow regime

V is given by

�(p) = �1

4�

ZZSb

�

rdS +

1

4�

ZZSb+Sw

�n � r1

rdS + �1(p); (3.10)

with the freestream potential, de�ned as �1(p) = U1X + V1Y +W1Z, to satisfy the

condition at in�nity. In two dimensions equation (3.10) is written as [50]

�(p) =1

2�

ZSb

� ln rdS �1

2�

ZSb+Sw

�n � r(ln r) dS + �1(p): (3.11)

Solving the Laplace equation (3.1) has now been reduced to �nding an appropriate

singularity distribution over the boundaries so that the boundary conditions (3.2) and

(3.3) are satis�ed. It is noted, however, that equations (3.10) and (3.11) do not imme-

diately specify a unique combination of sources and doublets for a particular problem.

Additional considerations are usually required which depend on the physics of the prob-

lem (i.e. Kutta condition to de�ne the ow near sharp trailing edges) to relate � in the

wake to � on the body.


3.2.2 Model problem

From the previous section it is clear that a solution for the ow over arbitrary bodies can

be obtained by a singularity distribution on the modelled surface. Before applying the

described method to practical problems, the method is �rst investigated for the model

problem of two- and three-dimensional non-lifting small disturbance ow over dented

plates, which corresponds with non-lifting thin-aerofoil/wing theory.

Let the surface of the dented plate be given by Z = Zp(X; Y ), with Zp assumed to be

small compared to the length and the width of the plate. In order to �nd the normal to

the surface, the function F (X; Y; Z) is de�ned

F (X; Y; Z) � Z � Zp(X; Y ) = 0: (3.12)

The unit normal vector n, pointing outward on the upper surface of the dented plate is

determined by

n =rFjrF j

=1

jrF j

��@Zp

@X;�

@Zp

@Y; 1

�T

: (3.13)

The velocity potential due to a freestream Q1 = (U1; V1; 0)T , corresponds with �1 =

U1X + V1Y , and the potential can be constructed to be

� = �+ �1; (3.14)

which has to ful�l the boundary condition (3.2) of zero normal velocity at the surface of

the plate

r� � n =�1jrF j

�@�

@X+ U1

�@Zp

@X+

�@�

@Y+ V1

�@Zp

@Y�

@�

@Z

!= 0: (3.15)

Consequently, the Laplace problem r2� = 0 is to be solved for the perturbation potential

� with boundary condition (3.15) for @�=@Z on Z = Zp. Assuming the geometry of the

dent to be shallow, the perturbation velocity can be taken to be small compared to the

freestream �� @�@X�� ;�� @�@Y

�� ;�� @�@Z

�� Q1; (3.16)

where Q1 = jQ1j and boundary condition (3.15) reduces to

@�

@Z(X; Y; Z = Zp) = U1

@Zp

@X+ V1

@Zp

@Y: (3.17)

With the use of a Taylor expansion, boundary condition (3.17) at the plate's surface is

transferred to the (X; Y ) plane (Z = 0)

@�

@Z(X; Y; 0) = U1

@Zp

@X+ V1

@Zp

@Y: (3.18)


The above de�ned potential- ow problem for the perturbation potential � and boundary

condition (3.18) can be solved by a source distribution on the (X; Y ) plane. Assuming

a source distribution of strength � on the (X; Y ) plane over the region X 2 [0;1] and

Y 2 [0;1], the velocity potential and velocity �eld are given by

�(X; Y; Z) =�14�

1Z0

1Z0

�(�; �) d�d�

((X � �)2 + (Y � �)2 + Z2)12

; (3.19)

@�

@X=

1

4�

1Z0

1Z0

�(�; �)(X � �) d�d�

((X � �)2 + (Y � �)2 + Z2)32

; (3.20)

@�

@Y=

1

4�

1Z0

1Z0

�(�; �)(Y � �) d�d�

((X � �)2 + (Y � �)2 + Z2)32

; (3.21)

@�

@Z=

1

4�

1Z0

1Z0

�(�; �)Z d�d�

((X � �)2 + (Y � �)2 + Z2)32

: (3.22)

The normal velocity @�=@Z(X; Y; 0) at the surface is related to the source strength,

see �gure 3.2. This can be shown either by a limit process for Z ! 0� in equation (3.22),

0+ and 0� representing the upper and lower surface, respectively, or by observing the

ux over the area �X by �Y . The ux due to a source of strength �(X; Y ) over the area

�X by �Y is given by

�(X; Y )�X�Y: (3.23)

At the same time, the ux due to the ow rate can be found to be

@�

@Z(X; Y; 0+)�X�Y �

@�

@Z(X; Y; 0�)�X�Y = 2

@�

@Z(X; Y; 0+)�X�Y; (3.24)

the uxes from the side becoming negligible when �Z ! 0. Modelling the ow over the

upper surface of the plate, only 0+ is of interest and�@�=@Z(X; Y; 0�)= @�=@Z(X; Y; 0+)

p

σ (X,Y)

ddZ

(X,Y,0+)

X

ddZ

(X,Y,0-)

ZY

Z

0

Q

V

U

ϕ

ϕ

∞

∞

∞

Figure 3.2: Source distribution for a dented plate.


= @�=@Z(X; Y; 0). The two uxes (3.23) and (3.24) should be the same and

�(X; Y ) = 2@�

@Z(X; Y; 0): (3.25)

Comparing (3.25) with boundary condition (3.18), it is seen that modelling the perturba-

tion potential over a dented plate corresponds with a source distribution over the (X; Y )

plane when

�(X; Y ) = 2

�U1

@Zp

@X+ V1

@Zp

@Y

�: (3.26)

The velocity �eld at the surface of the dented plate, Zp being small, is now given by

U(X; Y; Z = Zp) � U(X; Y; 0) =1

2�

1Z0

1Z0

W (�; �; 0)(X � �) d�d�

((X � �)2 + (Y � �)2)32

+ U1; (3.27)

V (X; Y; Z = Zp) � V (X; Y; 0) =1

2�

1Z0

1Z0

W (�; �; 0)(Y � �) d�d�

((X � �)2 + (Y � �)2)32

+ V1; (3.28)

where W (X; Y; 0) = @�=@Z(X; Y; 0) as de�ned in (3.18). In two dimensions a similar

result is found [50]

U(X;Z = Zp) � U(X; 0) =1

�

1Z0

W (�; 0)d�

X � �+ U1; (3.29)

where

W (X; 0) =@�

@Z(X; 0) = U1

@Zp

@X: (3.30)

3.2.3 Wing/aerofoil problem

In the previous section 3.2.2 the modelling of the inviscid ow over dented plates was

discussed with the boundary condition applied on the projection of the dent in the (X; Y )

plane (Z = 0). In order to more realistically model the geometry of a wing or aerofoil the

boundary conditions should be applied at the actual surface. The singularity elements

are to be distributed over the real surface to model the whole ow �eld, and the problem

is reduced to �nding the source and doublet strengths so that the boundary conditions

(3.2) and (3.3) are satis�ed.

Boundary condition (3.3) is automatically ful�lled as the singularity solutions have a

velocity �eld that decays for r ! 1. For boundary conditions (3.2) two formulations

exist. If the boundary-value problem is solved explicitly using boundary condition (3.2)

we speak of a Neumann problem, and equations (3.10) and (3.11) are di�erentiated in

order to solve for the total velocity �eld Qtot

Qtot = r�: (3.31)


Instead of applying the direct boundary condition (3.2), it is also possible to de�ne � as

a constant. This is termed the Dirichlet formulation of the problem.

For the work described in this thesis the Dirichlet method is applied on the wing's

surface for the modelling of the really inviscid wing ow. The solution in the wake

region follows, once the Dirichlet problem on the body is solved. The modelling of both

regions is discussed next in more detail. However, �rst the Kutta condition is de�ned

to determine the problem uniquely by relating the doublet strengths in the wake to the

doublet strengths at the trailing edge.

Kutta condition

In the case of modelling wing or aerofoil ow the problem is not uniquely speci�ed by

only the boundary condition of no normal velocity through the pro�le's surface.

A condition is required at the trailing edge to relate the doublets in the wake to the

body doublets in order to uniquely determine the solution. The Kutta condition is used

for this and in its most general form requires that the velocity at the trailing edge is

bounded

jr�jte < 1: (3.32)

For the present formulation an implicit Kutta condition is used. The potential jump

across the wake is set equal to the di�erence between the potential values at the upper

and lower surface at the trailing edge. This is the same as relating the doublet strength in

the wake �w to the unknown doublets �teu and �tel at the upper and lower trailing edge

of the wing's body as

�w = �teu � �tel: (3.33)

Dirichlet method

Specifying on the boundary the potential inside the body surface �in, instead of using

(3.2), is called the Dirichlet boundary condition and can be employed to determine the

solution of the boundary-value problem (3.1). For �eld points p inside the body, the

internal potential is given by (3.10)

�in = �1

4�

ZZSb

�

rdS +

1

4�

ZZSb+Sw

�@

@n

1

rdS + �1: (3.34)

With the Dirichlet boundary condition, �in = �in + �1 = �1, where the internal

perturbation potential �in is set to zero, equation (3.34) simply becomes

1

4�

ZZSb+Sw

�@

@n

1

rdS =

1

4�

ZZSb

�

rdS; (3.35)

which for the two-dimensional ow case corresponds with

1

2�

ZSb+Sw

�@

@n(ln r) dS =

1

2�

ZSb

� ln r dS: (3.36)


From the freestream the source strength � can be determined, using the de�nition of

source strength (3.9)

� �@�

@n�@�in

@n= 0�

@�1

@n= �Q1 � n; (3.37)

with �in = �1 and the normal velocity @�=@n at the surface being zero.

When applying the implicit Kutta condition (3.33), linking the doublet strengths in the

wake to the upper and lower trailing edge doublets, the above constant-potential Dirichlet

problem is uniquely de�ned and the unknown doublet distribution can be determined.

Assuming the wake to have zero thickness, the velocity distribution everywhere within

the ow regime V is automatically determined by the source and doublet distribution

provided by the Dirichlet method on the pro�le's surface. Equations (3.10) and (3.11),

di�erentiated in the normal direction with respect to the �eld point p, form the integral

equations for the velocity �eld in the wake and for the three-dimensional ow problem

given by equation (3.10) this results in

Qtot(p) = r�(p)

= r��

1

4�

ZZSb

�

rdS +

1

4�

ZZSb+Sw

�@

@n

1

rdS + �1(p)

�: (3.38)

3.2.4 Full-potential problem

As part of this thesis, contract work has been carried out for the Defence Evaluation and

Research Agency (DERA) in Farnborough, United Kingdom, and use has been made of

their existing Viscous Full-Potential (VFP) code for transonic ow calculations.

The inviscid ow part of the VFP program is based on the method developed by Forsey

and Carr [36] for calculating potential ow over wing-fuselage combinations. It is valid

for non-zero, subsonic Mach numbers and the scheme can be applied to con�gurations

with swept, cranked and tapered wings. As the full-potential (FP) part of the VFP code

is treated as a blackbox it will not be discussed in this thesis. More information can be

found in the papers [33, 34, 4].

3.3 Viscous potential- ow problem

In the previous sections the modelling of the inviscid external ow has been discussed,

ignoring the presence of the boundary layer. However, as the boundary layer displaces

the outer potential ow away from the surface over a distance, termed the displacement

thickness, this boundary-layer displacement e�ect should be taken into account into the

external ow modelling.

The potential ow should be modelled over the resulting e�ective body shape, con-

structed from the actual physical body plus the displacement thickness. This can be done

via the distribution of singularities over the resulting e�ective body shape which forms a

streamsurface [60]. A simpler approach is to change the zero normal velocity boundary

condition at the original surface in such a way that the normal velocity at the real surface

3.3 Viscous potential- ow problem 37

in the potential ow corresponds with the normal velocity at the edge of the boundary

layer.

Before continuing, a local Cartesian coordinate system (x; y; z) is de�ned where the

x- and y-axis are tangent to the surface and the z-axis is normal to the surface. The

local total velocity vector is given by qtot = (u; v; w; )T . The velocity at the edge of the

boundary layer is indicated with the subscript e, thus ue and ve are the x- and y-velocity

components at the edge of the boundary layer in the local Cartesian system.

Using the continuity equation and assuming that w grows linearly with � (@ue=@x+

@ve=@y) z within the boundary layer, the following approximation for the normal velocity

in the boundary layer can be obtained:

w(x; y; z) � ��@ue

@x+@ve

@y

�z + w1(x; y); z !1; (3.39)

where z !1 indicates the edge of the boundary layer. The disturbance velocity w1 6= 0 is

the next term of the expansion of w for z !1 and represents the so-called transpiration

velocity, describing the transpiration between the boundary layer and the external ow.

The de�nition for the transpiration velocity w1 is determined by

w1(x; y) = limz!1

[w(x; y; z) +@ue

@xz +

@ve

@yz]

= limz!1

[

zZ0

@w

@zdz +

@ue

@xz +

@ve

@yz]

= limz!1

[

zZ0

��@u

@x�@v

@y

�dz +

@ue

@xz +

@ve

@yz]

= limz!1

[

zZ0

�@ue

@x�@u

@x

�dz +

zZ0

�@ve

@y�@v

@y

�dz]

=@

@x

1Z0

(ue � u)dz +@

@y

1Z0

(ve � v)dz

=@

@x(qeÆ

�

x) +

@

@y(qeÆ

�

y); (3.40)

where qe =pu2e+ v2

e+ w2

eis the velocity magnitude at the edge of the boundary layer.

The integral thickness de�nitions in the local Cartesian coordinate system for the dis-

placement thicknesses Æ�xand Æ�

yin the x- and y-directions can be found in appendix

A.

Physically, the above transpiration velocity represents the rate of change of mass

defect, being mx = qeÆ�

xand my = qeÆ

�

yin the local Cartesian coordinate system.

In two dimensions equation (3.40) reduces to

w1(x) =@

@x(qeÆ

�

x): (3.41)


To include the viscous in uence in the potential- ow problem described in the previ-

ous sections, the `zero' normal velocity boundary condition at the body surface (3.2) is

modi�ed, giving

r� � n = w1(x; y) 6= 0: (3.42)

For the discussed potential- ow problems the change in boundary condition, due to the

presence of the boundary layer, results in modifying the source strengths. For the three-

dimensional dented plate problem, where the global coordinate system corresponds with

the local coordinate system, the source strength (3.26) becomes

�3D = 2

�@

@x(U1Zp) +

@

@y(V1Zp)

�+ 2

�@

@x(qeÆ

�

x) +

@

@y(qeÆ

�

y)

�= �inv + �vis; (3.43)

or in two dimensions

�2D = 2@

@x(U1Zp) + 2

@

@x(qeÆ

�

x) = �inv + �vis; (3.44)

where �inv represents the inviscid source strength based on the freestream and �vis rep-

resents the viscous source strength due to the presence of the boundary layer.

For the wing and aerofoil problem the viscous sources should be included on the body

and in the wake, and equation (3.37) for the source distribution on the body surface

changes to

� = �Q1 � n+ w1 = �inv + �vis: (3.45)

In the wake, the displacement e�ect is a jump in normal velocity �w1 across the wake

centre line, dividing the ows coming from the upper and lower surfaces of the pro�le

�w1 =@

@x(qeuÆ

�

xu+ qelÆ

�

xl) +

@

@y(qeuÆ

�

yu+ qelÆ

�

yl); (3.46)

where subscripts u and l indicate upper and lower surface values, respectively.

Dependent on the modelling of the boundary layer for the various ow cases, the

wake's upper and lower layers will be treated separately or together as just one layer. In

the �rst case the above boundary condition (3.46) on the wake dividing streamsurface

leads to the following viscous source strength in the wake:

� =@

@x(qeuÆ

�

xu+ qelÆ

�

xl) +

@

@y(qeuÆ

�

yu+ qelÆ

�

yl) = �vis: (3.47)

For the case where the wake is modelled as just one layer with the wake thicknesses de�ned

with Æ�xw

= Æ�xu+ Æ�

xland Æ�

yw= Æ�

yu+ Æ�

yl, and the velocity qe = qeu = qel, the viscous source

strength becomes

�vis =@

@x(qeÆ

�

xw) +

@

@y(qeÆ

�

yw): (3.48)

3.4 Discretisation viscous potential- ow problem 39

The general formulation of the two-dimensional potential- ow problem in (3.11) with

the inclusion of viscous e�ects can be written

�(p) =1

2�

ZSb

�inv ln r dS �1

2�

ZSb+Sw

(�n � r(ln r)� �vis ln r) dS + �1(p); (3.49)

with the inviscid and the viscous source strengths on the body de�ned in (3.45). The

wake is modelled as just one layer, implying that �vis in the wake is given by the two-

dimensional version of (3.48).

3.4 Discretisation viscous potential- ow problem

For the discretisation of the potential-based equations use is made of a panel method.

The body surface, and (if present) the wake surface, are divided into a number of panel

elements as seen in �gure 3.3. For the three-dimensional case planar panels are used,

whereas in two dimensions the panels simply reduce to straight lines.

The discretisation for the viscous potential problem for indented plate ow and wing/ae-

rofoil ow is examined, and the velocity equations at the edge of the boundary layer

required for the coupled viscous/inviscid problem are determined.

For clarity, �rst the used local and global Cartesian coordinate systems are re-introduced.

3.4.1 Global and local coordinate systems

In section 3.2 the general formulation of the potential- ow problem has been described

in a global body-�xed Cartesian coordinate system (X; Y; Z) with velocity vector Qtot =

(U; V;W )T . When the geometry is discretised, it is useful to de�ne for each panel segment

a local panel Cartesian coordinate system (x; y; z) with velocity vector qtot = (u; v; w)T .

For the model problem of ow over a dented plate, the local and global Cartesian

coordinate systems are chosen to correspond with each other, with the origin de�ned

at a corner point of the plate. However, for the panel method applied for a wing or

aerofoil section the panels no longer lie in the same plane. Each panel has a di�erent local

trailing edge

Z

Yglobal

tip

panel

z y

x

local

individual

W I N G

root

leading edge

Q∞

X

Figure 3.3: Global (X; Y; Z) and local (x; y; z) Cartesian coordinate systems.


coordinate system, with the origin chosen in the midpoint of each panel, see �gure 3.3.

The z-axis is taken to be normal to the panel and the x- and y-axis are tangent to the

panel.

3.4.2 Discretisation two-dimensional model problem

A two-dimensional plate is de�ned between x1 and xMp+1 along the x-axis and integral

(3.29), which represents the x-velocity component at the edge of the boundary layer,

becomes

ue(x) =1

�

xMp+1Zx1

we(�)d�

x� �+ u1(x); (3.50)

where 2we = �inv + �vis as given in equation (3.44). For the discretisation of the above

integral, segment [x1; xMp+1] is divided into Mp straight-line panels each of similar length

�x = xk+1�xk for k 2 [1;Mp]. The collocation points are de�ned at the panel endpoints

x = xi for i 2 [2;Mp] and

uei =1

�

xMp+1Zx1

we(�)d�

xi � �+ u1i

: (3.51)

It is seen that for � = xi the above integral is singular and in the region surrounding the

singularity [xi�1; xi+1] a linear expansion Lwe of we is to be used. The Cauchy principal

part (C:p:p:) of the singular integral can be calculated and (3.51) is

uei =1

�

MpXk=1

k 6=i�1;i

xk+1Zxk

wek+1

2

d�

xi � �+

1

�

xi+1Zxi�1

Lweid�

xi � �+ u1i

=1

�

MpXk=1

k 6=i�1;i

wek+1

2

ln

�� xi � xk

xi � xk+1

��2�x

�

@we

@x

��i| {z }

C:p:p:

+u1i: (3.52)

where

Lwei= wei

+ (� � xi)@we

@�

��i

=@

@�(u1Zp + qeÆ

�

x)ji + (� � xi)

@2

@�2(u1Zp + qeÆ

�

x)ji: (3.53)

Using the above de�nition of Lwe and centrally discretising the gradients of u1Zp and

qeÆ�

x, equation (3.52) becomes

uei =

MpXk=1

k 6=i�1;i

qek+1Æ�

xk+1� qekÆ

�

xk

��xln

�� i� k

i� k � 1

�� 2qei+1Æ

�

xi+1� 2qeiÆ

�

xi+ qei�1Æ

�

xi�1

��x+ u0ei ;


which can be rewritten to the form

uei =

Mp+1Xk=1

Au

ikqekÆ

�

xk+ u0ei ; i = 2; : : : ;Mp; (3.54)

where u0e is the undisturbed external inviscid ow at the edge of the boundary layer due

to the freestream and the geometry of the dent. The summation part on the right-hand

side of equation (3.54) describes the disturbance e�ect due to the presence of the boundary

layer, with matrix Au constructed as follows:

Au

ii=

4

��x; i = 2; : : : ;Mp;

Au

ii�1 = �1

��x(2� ln 2); i 6= 2;

Au

ii+1 = �1

��x(2� ln 2); i 6=Mp;

Au

ik=

1

��xln

��1� 1

(i� k)2

��; k 6= 1; i� 1; i; i+ 1;Mp;

Au

21 = �2

��x;

Au

i1 = �1

��xln

i

i� 1; i 6= 2;

Au

iMp+1= �

1

��xlnMp + 1� i

Mp � i; i 6=Mp;

Au

MpMp+1= �

2

��x:

The constructed in uence matrix Au has the following characteristics:

Au

ii> 0;

Au

ik= Au

ki< 0; (i 6= k), (3.55)

Mp+1Xk=1

Au

ik� 0; with at least for one i the inequality sign.

From (3.55) it follows, that matrix Au is a symmetric weakly diagonally dominant M-

matrix (de�nition D.3 and theorem D.10) and therefore positive de�nite (theorem D.12).

3.4.3 Discretisation three-dimensional model problem

In the Cartesian coordinate system a rectangular plate is de�ned between x1 and xMp+1

and between y1 and yMs+1 in the (x; y) plane. The segments along the x-axis are de�ned

by the intervals [xk; xk+1] of length �x for k 2 [1;Mp] and along the y-axis by [yl; yl+1] of

length �y for l 2 [1;Ms]. The grid is constructed by rectangulars, resulting in panels of

�x by �y. The double integrals are evaluated in the node points of the panels (xi; yj),


for i = 2; : : : ;Mp and j = 2; : : : ;Ms and the velocity components (3.27) and (3.28) at

the edge of the boundary layer become

ueij =1

2�

xMp+1Zx1

yMs+1Zy1

we(�; �)(xi � �)d�d�

((xi � �)2 + (yj � �)2)32

+ u1ij; (3.56)

veij =1

2�

xMp+1Zx1

yMs+1Zy1

we(�; �)(yj � �)d�d�

((xi � �)2 + (yj � �)2)32

+ v1ij: (3.57)

It is seen that the above integrals, when evaluated in � = xi and � = yj are singular.

Outside a region surrounding the singularity, the integrals are well-behaved. As in the

two-dimensional case it is assumed that a linear source distribution exists over the region

xi�1 < x < xi+1 and yj�1 < y < yj+1, including the singularity. The Cauchy principal

part of the singular integral can be calculated and equation (3.56) is rewritten

ueij =1

2�

MpXk=1

k 6=i�1;i

MsXl=1

l6=j�1;j

xk+1Zxk

yl+1Zyl

wek+1

2l+1

2

(xi � �)d�d�

((xi � �)2 + (yj � �)2)32

+1

2�

xi+1Zxi�1

yj+1Zyj�1

Lweij(xi � �)d�d�

((xi � �)2 + (yj � �)2)32

+ u1ij

=1

2�

MpXk=1

k 6=i�1;i

MsXl=1

l6=j�1;j

wek+1

2l+1

2

ln

��p(xi � xk+1)2 + (yj � yl)2 + (yj � yl)p

(xi � xk+1)2 + (yj � yl+1)2 + (yj � yl+1)

��+ ln

��p(xi � xk)2 + (yj � yl+1)2 + (yj � yl+1)p(xi � xk)2 + (yj � yl)2 + (yj � yl)

��!

��y

2�ln

��(p�x2 +�y2 +�x)2

(p�x2 +�y2 ��x)2

�� @we

@x

��ij| {z }

C:p:p:

+u1ij; (3.58)

with Lwe the linear expansion of we

Lweij= weij

+ (� � xi)@we

@�

��ij

+ (� � yj)@we

@�

��ij

; (3.59)

and 2we = �inv + �vis given by (3.43) is

we =@

@x(u1Zp + qeÆ

�

x) +

@

@y(v1Zp + qeÆ

�

y): (3.60)

Using the de�nition of Lwe (3.59) and centrally discretising the gradients of qeÆ�

x, qeÆ

�

y,


u1Zp and v1Zp, the equations for ueij and veij are found to be of the form

ueij =

Mp+1Xk=1

Ms+1Xl=1

qekl�Au

ijklÆ�xkl

+Bu

ijklÆ�ykl

�+ u0eij ; (3.61)

veij =

Mp+1Xk=1

Ms+1Xl=1

qekl�Av

ijklÆ�xkl

+Bv

ijklÆ�ykl

�+ v0e

ij: (3.62)

The double summation part contains the disturbance due to the boundary layer, and

as before, u0e and v0e are the undisturbed inviscid velocity components, containing the

freestream and the discretisation of the geometry.

3.4.4 Discretisation wing/aerofoil problem

The Dirichlet approach of the potential- ow problem is to be applied for the wing/aerofoil

problem. With the inclusion of the boundary-layer displacement e�ect in the Dirichlet

formulation of the potential- ow problem on the wing's surface, equation (3.35) becomes

1

4�

ZZSb+Sw

�@

@n

1

rdS =

1

4�

ZZSb

�inv

rdS +

1

4�

ZZSb+Sw

�vis

rdS; (3.63)

with the inviscid source strength �inv and the viscous source strength �vis as de�ned in

(3.45) on the body and (3.47) in the wake.

In order to calculate the above integrals the wing surface and wake are discretised.

Let the number of segments along the contour of the body surface be denoted by Mb, and

spanwise along the Y -axis by Ms, resulting in Nb = Mb�Ms panels on the body surface.

The wake centre line is divided into Mw segments, leading to Nw = Mw �Ms panels in

the wake.

A low-order panel method is used, assuming that the singularity strengths are constant

and equal to the value at the centroid of each panel. Each panel will have a constant

doublet strength �n and a constant source strength �n, for n 2 [1; Nb+Nw]. The inviscid

source strengths are determined from the freestream and the viscous source strengths are

known from the viscous quantities. The discretisation of equation (3.63) then leads to a

linear system of equations to be solved for the unknowns �n, n 2 [1; Nb +Nw], de�ned in

the panel centres

Nb+NwXn=1

1

4�

ZZSn

�n@

@n

1

rdS =

NbXn=1

1

4�

ZZSn

�invnr

dS +

Nb+NwXn=1

1

4�

ZZSn

�visnr

dS:

A more simpli�ed way of writing the above system for body collocation pointsm 2 [1; Nb],

located in the midpoint of the panel, is

Nb+NwXn=1

Amn�n =

NbXn=1

Bmn�invn +

Nb+NwXn=1

Bmn�visn ; (3.64)


and a similar system is found for the two-dimensional ow problem. For the construction

of (3.64) the source and doublet strengths, assumed constant per panel, have been taken

outside the double integral. In uence matrices Amn and Bmn, describing the in uence at

collocation point m due to a constant doublet strength distribution and source strength

distribution, respectively, placed at panel n, are given by

Three-dimensional case Two-dimensional case

Amn =1

4�

ZZSn

@

@n

1

rmn

dS; Amn =1

2�

ZSn

@

@n(ln rmn) dS; (3.65)

Bmn =1

4�

ZZSn

1

rmn

dS; Bmn =1

2�

ZSn

ln rmn dS: (3.66)

with rmn the distance between collocation point m and the point of integration n. The

matrices Amn and Bmn are de�ned in a local panel-orientated coordinate system, and the

evaluation of the above integrals can be found in the paper of Hess and Smith [44].

The above system (3.64) consists of Nb equations, however, there are Nb+Nw unknown

doublet strengths to be determined. For the problem to be unique the implicit Kutta

condition is used, to prescribe the unknown doublet strength in the wake. Providing

there is no signi�cant ow along the trailing edge, the doublet strength along a wake strip

is taken to be constant, and its value, �w, is related to the upper and lower trailing edge

doublets as

�w = �bu � �bl; (3.67)

where �bu is the doublet strength of the upper body trailing edge panel and �bl is the

doublet strength of the lower body trailing edge panel. With the above implicit Kutta

condition, the in uence matrix Amn can be reduced to a Nb�Nb matrix. The Nb unknown

doublet strengths on the body surface are to be calculated from the resulting system of

Nb equations.

Edge velocity equations body

For the interaction with the viscous layer, a relation between the velocity from the external

ow at the edge of the boundary layer, and the boundary-layer displacement thicknesses,

is required. For the Dirichlet method this means that equation (3.64) has to be further

manipulated.

The total edge velocity vector Qtote , located at the endpoint of a panel, instead of at

the centroid, can be decomposed as given in the paper by Kerwin et al. [51] and shown

in �gure 3.4

Qtote = Qse +Q1 + � n

= Qse +Q1 � (Q1 � n)n+ w1n; (3.68)

and ful�ls the boundary condition Qtote �n = w1 as given in (3.42). The term Qse = rs�

corresponds to the component of perturbation velocity tangential to the body surface. If


��

��

QQ

Qs

Q

n

n Q∞ w1

n-.-( )

∞

tot

Figure 3.4: Composition velocity vectors.

t1 and t2 are de�ned as two di�erent unit vectors tangential to the surface, the following

holds:

@�

@t1= t1 � rs�;

@�

@t2= t2 � rs�:

When using the above, the tangential velocity Qse can be expressed

Qse =

@�

@t1[ t1 � (t1 � t2) t2 ] +

@�

@t2[ t2 � (t1 � t2) t1 ]

k t1 � t2 k2: (3.69)

The two perturbation velocity components in the t1 and t2 directions are determined by

the two components of the doublet gradient in the two corresponding tangential directions

@�

@t1=

@�

@t1;

@�

@t2=

@�

@t2;

which can be discretised using central di�erencing.

After some manipulations, combining (3.64) with the Kutta condition (3.67) and

(3.69), the equation for Qse in the panel endpoints indicated by (i; j), with i 2 [2;Mb]

and j 2 [2;Ms], can be written

Qseij=

MbXk=1

MsXl=1

(CA�1B )ijkl �invkl +

Mb+MwXk=1

MsXl=1

(CA�1B )ijkl �viskl ; (3.70)

where indices k 2 [1;Mb] and l 2 [1;Ms] de�ne the panel number. Furthermore, matrix

(CA�1B) is constructed from three matrices A�1, B and C. Matrix A�1 is the inverse of

A which represents the doublet in uence and matrix B represents the source in uence,

both given in (3.65) and (3.66). Matrix C contains the discretisation of expression (3.69).

It follows that equation (3.68) for the total edge velocity Qtote , with Qse given by (3.70),

can be rewritten as:

Qtoteij= �visijn +

Mb+MwXk=1

MsXl=1

Dijkl �viskl +Q0eij; (3.71)


with matrix D being (CA�1B). In the above expression, Q0e is the undisturbed inviscid

velocity, due to the freestream and the inviscid sources.

Equation (3.71) can be further simpli�ed, using (3.45) and (3.47) for a two-layered

wake and centrally discretising the gradients of qeÆ�

xand qeÆ

�

y, giving �nally for the global

edge velocity components Qtote = (Ue; Ve;We)T in the body collocation points i 2 [2;Mb]

and j 2 [2;Ms]

Ueij=

Mb+2Mw+1Xk=1

Ms+1Xl=1

qekl [AU

ijklÆ�xkl

+BU

ijklÆ�ykl] + U0eij

; (3.72)

Veij =

Mb+2Mw+1Xk=1

Ms+1Xl=1

qekl [AV

ijklÆ�xkl

+BV

ijklÆ�ykl] + V0eij ; (3.73)

Weij=

Mb+2Mw+1Xk=1

Ms+1Xl=1

qekl [AW

ijklÆ�xkl

+BW

ijklÆ�ykl] +W0eij

: (3.74)

It is noted that the �rst wake node point corresponds with the upper and lower trailing

edge points, de�ned (Mw + 1; l) and (Mb +Mw + 1; l), and is therefore omitted. Indices

(Mw; l) and (Mb+Mw +2; l) thus represent the upper and lower `second' wake point just

after the trailing edge for spanwise station l.

In two dimensions a similar manipulation can be followed and relations for the velocity

components U and W in the Cartesian coordinate system can be found. However, un-

like the three-dimensional ow problem where the velocities in the Cartesian coordinate

system are used, the two-dimensional ow problem will make use of a streamline coordi-

nate system (s; n). The tangential velocity in the streamline direction Us =pU2 +W 2

becomes

Usei= Qtotei

� sji

=@�

@s+@�1

@s; (3.75)

having used (3.68) with Qse � s = @�=@s = @�=@s, � n � s = 0 and Q1 � s = @�1=@s. By

manipulating (3.64) and using the Kutta condition (3.67) it is then found that

Usei=

MbXk=1

(CA�1B )ik �invk +

Mb+MwXk=1

(CA�1B )ik �visk + Us1i; (3.76)

with matrix C containing the discretisation of (3.75). Modelling the boundary layer in

the wake as just one layer with (3.48) and using the integral thickness de�nitions in a

streamline coordinate system, as given in appendix A, Use becomes

Usei=

Mb+Mw+1Xk=1

AUs

ikUse

kÆ�sk+ U0sei

: (3.77)

From equations (3.72) and (3.74) it follows that:

AUs

ik= cos �k(cos �iA

U

ik+ sin �iA

W

ik); (3.78)

where �k is the angle panel k makes with the X-axis.

3.5 Aerodynamic coeÆcients 47

Edge velocity equations wake

In the wake region, the doublet strengths are known from the Dirichlet method on the

body surface and the implicit Kutta condition. The source strengths are given by the

freestream and viscous quantities.

For the discretisation of the potential- ow method in the wake, the potentials are to

be determined in the midpoints of the panels. By taking the gradient, the edge velocity

equations are calculated back in the node points of the panels. The total potential at the

collocation points m 2 [1; Nw], located in the midpoints of the wake panels, is given by

�m = �1m�Nb+NwXn=1

1

4�

ZZSn

�n@

@n

1

rdS +

NbXn=1

1

4�

ZZSn

�invnr

dS +

Nb+NwXn=1

1

4�

ZZSn

�visnr

dS:

A more simpli�ed way of writing the above system is

�m = �1m�

Nb+NwXn=1

Amn�n +

NbXn=1

Bmn�invn +

Nb+NwXn=1

Bmn�visn (3.79)

= �1m+ �m;

with Amn and Bmn as de�ned before in (3.65) and (3.66) and � the perturbation potential.

The doublet strengths � in (3.79) are known from the Dirichlet method on the body and

can be replaced with (3.64), which results in an equation for � only dependent on the

sources strengths �inv and �vis.

The velocity distribution Qtote can then be determined in the panel endpoints with

index (i; j), i 2 [1;Mw � 1] and j 2 [2;Ms], by di�erencing, using (3.69). Note that

i = 1 represents the `second' wake point after the trailing edge. Relations for the velocity

components in the wake are thus found of a similar format as equations (3.72), (3.73) and

(3.74).

3.5 Aerodynamic coeÆcients

In a frame of reference with velocity Q1, Bernoulli's equation for incompressible and

inviscid steady ow is

p1 + 12�Q2

1= p+ 1

2�Q2

tot; (3.80)

where p is the pressure and � = 1 the density. From (3.80) the pressure coeÆcient follows:

Cp =p� p112�Q2

1

= 1�Q2tot

Q21

: (3.81)

In two dimensions the lift and the drag coeÆcient are de�ned by

CL =L

12�Q2

1c; CD =

D12�Q2

1c; (3.82)


for which the lift L, is the force acting in the direction perpendicular to the freestream

direction and the drag D, the force acting in the direction of the freestream.

Assuming the pressure coeÆcient Cp and the skin-friction coeÆcient Cf to be constant

over each panel, the forces in the X- and Z- direction per wing section are given by

Fx =Xi

(Cpili sin �i + sign

Cfi

2li cos �i); (3.83)

Fz =Xi

(�Cpili cos �i + sign

Cfi

2li sin �i); (3.84)

where �i the angle between panel i and the X-axis and l2i= �X2 + �Z2, the length of

panel i. The term sign is equivalent to +1 or �1 for panel i on the upper or lower surface,respectively. Then for an incidence � the lift and drag are

L = Fz cos�� Fx sin�; (3.85)

D = Fx cos�+ Fz sin�: (3.86)

Alternatively, the lift coeÆcient per wing section can be determined with the Kutta-

Joukowski's formula [50]. This formula states that the resultant aerodynamic force in

an incompressible, inviscid, steady and irrotational ow is directly proportional to the

circulation � and acts normal to the freestream, hence

CLKJ=

�12�Q2

1c: (3.87)

The total circulation is determined by

� = �wu � �wl= �teu � �tel;

having used the implicit Kutta condition (3.33).

The integration formula for the drag coeÆcient (3.86) can give quite incorrect results.

A more suitable prediction is given with Young's formula [109]

CDY=

2�ss1c

= �sste�Usete

=Q1�(Hte+5)=2

; (3.88)

which is derived from the Von K�arm�an equation, with �ss1 the streamwise momentum

thickness far downstream in the wake.

3.6 Veri�cation two-dimensional Dirichlet method

To verify the two-dimensional Dirichlet method for the calculation of inviscid ow, the

described panel method has been tested for the calculation of inviscid ow around an

3.6 Veri�cation two-dimensional Dirichlet method 49

ellipse for which the analytical solution is known. The geometry of the ellipse is de�ned

by

(X � 12)2

(12)2

+Y 2

� 2= 1; (3.89)

where 2� is the maximum thickness of the ellipse. The maximum width of the ellipse

corresponds with 1. The exact solution for the ow past an ellipse can be determined

with complex-function theory and the velocity components in the X- and Y -direction on

the ellipse are

Uellipse =

�0:5� �

0:5 + �+ 1

�(2� 8(x� 0:5)2) cos� + (4y � 4yx)=� sin�

1 + 0:5��0:5+�

�0:5��0:5+�

� 16(x� 0:5)2 + 2� ; (3.90)

Vellipse =

�0:5� �

0:5 + �� 1

�(4x� 8(x� 0:5)2) sin� + (4y(x� 0:5))=� cos�

1 + 0:5��0:5+�

�0:5��0:5+�

� 16(x� 0:5)2 + 2� ; (3.91)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 0.2 0.4 0.6 0.8 1

Y

X

ELLIPS

Figure 3.5: Geometry ellipse for � = 0:06.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1

U_s_ellips

X

EXACT alpha = 0DIRICHLET alpha = 0



Figure 3.6: Velocity distribution for N = 15.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1

U_s_ellips

X





0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1

U_s_ellips

X






with � being the angle of attack. The total magnitude of the velocity corresponds with

the streamwise velocity at the surface of the ellipse and is given by

Usellipse=

qU2ellipse

+ V 2ellipse

: (3.92)

For the present calculations � is set to 0:06 for which the geometry of the ellipse is shown

in �gure 3.5. The trailing edge point corresponds with x = 1. The computations are

performed for three di�erent incidences, being � = 0o, 10o and 45o.

In �gures 3.6, 3.7 and 3.8 the velocity distribution obtained with the panel method

for the three incidences is shown, compared to the exact results obtained with (3.92). For

the results in �gure 3.6, 15 points are use for the discretisation of the body surface, and

it is seen that the results from the Dirichlet method are not very accurate. Increasing the

number of points to 35 gives much better correspondence with the exact solution, as is

seen in �gure 3.7. However, near the trailing edge the velocity distribution determined

by the panel method for the upper surface is seen to cross the velocity distribution for

the lower surface, resembling a �sh-tail. The analytical Kutta condition, implying equal

upper and lower velocities at the trailing edge, is not satis�ed.

The results obtained using 61 body points, shown in �gure 3.8, are nearly similar to

the results for 35 body points in �gure 3.7. The accuracy has not really improved and

the �sh-tail behaviour near the trailing edge for larger angles of attack remains. For the

boundary-layer coupling this can cause problems and a modi�cation of the trailing edge

values is required and will be discussed in the next section.

The three-dimensional Dirichlet method has been programmed by George Patrianakos,

a fellow Ph.D. student at the University of Bristol, and the validation for the three-

dimensional Dirichlet method will be presented in his thesis.

3.7 Adjustment trailing edge velocity distribution

As is clear from the previous section, the inviscid ow solution obtained at the trailing

edge with the two-dimensional Dirichlet panel method is not very accurate. The three-

dimensional wing Dirichlet panel method, making use of the same Kutta condition as the

two-dimensional method, displays the same inaccuracies.

With the implemented implicit Kutta condition (3.67) the actual Kutta condition

is not explicitly ful�lled as (3.67) apparently allows for the trailing edge velocities to

be di�erent from each other on the upper and lower surfaces, whereas analytically they

should be exactly the same. Already a small di�erence can have a large in uence on the

overall solution, especially when coupled to a boundary layer. An even larger problem

arises when the inaccurate solution at the trailing induces a steep pressure gradient, which

can cause the boundary-layer calculations to breakdown.

For the work described in this thesis the external ow solver is treated as a blackbox.

To avoid the breakdown of the boundary-layer calculations, and to possibly improve the

overall solution, therefore only some ad hoc modi�cations have been implemented. To

come to a more suitable trailing edge treatment the problem is to be examined more

closely.

3.7 Adjustment trailing edge velocity distribution 51

The velocities at the trailing edge coming from the wing and aerofoil Dirichlet panel

method are modi�ed before being used for the boundary-layer calculations. Various pos-

sibilities have been tested and in the two-dimensional aerofoil code the following simple

procedure is used:

Uete = min(Ueteu; Uete

l): (3.93)

The new value at the trailing edge is set to the minimum of the upper and lower trailing

edge velocities.

For a wing section at angle of attack the most critical area is near the trailing edge

on the upper surface, where separation can occur for severe positive pressure (negative

velocity) gradients. Adjustment by using the maximum value of Uete at the trailing

edge, instead of the minimum value as is done in (3.93), would decrease the velocity

gradient near the trailing edge of the upper surface and one would expect this to improve

the boundary-layer calculations. However, using the minimum has shown to work much

better. An explanation for this could be that the pressure gradient just after the trailing

edge in the wake is more problematic than the pressure gradient at the trailing edge on

the upper surface. Taking the maximum value of Uete at the trailing edge could in this

case increase the (positive) pressure gradient just after the trailing edge in the wake,

making the calculations there more diÆcult, whereas taking the minimum would make

the calculations easier.

For the three-dimensional wing external velocity distribution a slightly di�erent mod-

i�cation procedure is used. The solution for the three velocity components in the trailing

edge point on the upper and lower surface is determined by averaging the calculated upper

and lower trailing edge velocities.

In the wake the solution for the Z-velocity component in the �rst point downstream of

the trailing edge is also redetermined. This is done via interpolation with the new trailing

edge velocity and the downstream wake values.

Other adjustment variants have been tested as well, however, the above ones have

shown to be the most robust and accurate and have been applied.

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