Numerical Methods in Aircraft Aerodynamics, Part I: Panel Methods

Lecture Notes D-42

March 1993 Prof.ir. J.W. Slooff

Deift University of Technology

Numerical Methods in Aircraft

Aerodynamics

Part I: Panel Methods _{TECHNISCHE UNIVERSITEIT}

Laboratorium voor

Scheepshydromechan

Archief

Meketweg 2,2628

_{CD Dell!}

ToL 015.786ß73 - Fac 015-781838

Aerodynamics

Part I: Panel Methods

NUMERICAL METHODS IN AIRCRAFT AERODYNAMICS

CONTENTh Page

O. Preface O-1

1. INTRODUCTION AND OVERVIEW 1-1

1.1 Numerical Aerodynamics; What, Why and How? l-1

1.2 More on 'How' 1-3

1.2.1 Sketch of the Physical Problem l-3

1.2.2 Mathematical-Physical Models

i-k

1.2.3 Numerical Models 1-18

1.2.4 Relative Computational Efforts l-21

1.3 Aspects of Application 1-21

1.4 State-of-the-Art 1-23

2. PANEL METHODS _2-1

2.1 Mathematical Foundations 2-1

2.1.1 Basic Equations _2-1

2.1.2 Boundary Conditions and Additional Conditions; _2-5 Uniqueness of' the Boundary Value Problem

2.1.3 Modeling of Circulation/Lift 2-8

2.1.4 Summary of the Boundary Value Problem for Wing-Like bodies 2-20

2.1.5 Models for Other Types of Bodies _2-21

2.1.6 Elementary Solutions of Laplace's Equation 2-24

2.1.7 Green's identity _2-35

2.1.8 Boundary value problems and equivalent integral equations _2-37 2.1.9 General formulation of' boundary value problems in terms of

source and doublet distributions using Green's identity

2-39

2.1.10 Uniqueness of the integral representation derived from

the general formulation

2.1.12 Summary of well-posed and some ill-posed boundary value 2-45

problems and integral representations

2.1.13 The application of the integral representation to the 2-47

boundary value problem for the flow around an aircraft

configuration

2.2 Numerical Aspects 2-55

2.2.1 General considerations on the discretisation of the 2-55

integral equations

2.2.2 Surface-grid generation ('paneling') 2-66

2.2.3 Local representation of the geometry of a panel 2-68

2.2.4 Representation of source and doublet distributions 2-72 over a panel

2.2.5 Aerodynamic influence coefficients 2-7'-t

2.2.6 Truncation errors; consistent approximations 2-86

2.2.7 Convergence and stability 2-101

2.2.8 Factors affecting the magnitude of the truncation error 2-107

2.2.9 Solution methods for the resulting system of linear 2-108 algebraic equations; computational effort

2.2.10 Post-processing; lift and drag 2-116

2.3 The Modeling of Compressibility (Subsonic) 2-119

2.3.1 The classical Göthert rule 2-120

2.3.2 Application of the Göthert rule in panel methods 2-122 2.3.3 Alternative treatments, mass flux boundary condition 2-124

2.3.4 Summary of linearized models for subsonic 2-125

2.4 Examples of Application (3D) of 'Ordinary Panel Methods 2-129

2.4.1 Status arid role of panel methods in the process of 2-129

aircraft aerodynamic design

2.4.2 Low speed flow 2-129

2.4.3 High speed, subsonic flow 2-131

2.5 Simulation of viscous effects 2-132

2.6 Special Methods 2-136

2.6.1 Methods based on thin-wing theory 2-136

2.6.2 Panel methods for supersonic flow 2-140

2.6.3 Methods for modeling vortex sheet roll-up 2-142

2.6.4 Other methods 2-146

O. PREFACE

The purpose of these lecture notes is to make students in aeronautical

engineering of graduate or final-stage undergraduate level, specialising in aerodynamics, acquainted with the (relatively young) discipline of NUMERICAL

AERODYNAMICS.

More specifically it is intended to provide the student with knowledge that will enable, or help him/her to appreciate the possibilities and limitations of the numerical methods that are currently utilised in aerodynamic design and research

environments. For this purpose the course attempts to provide a balanced

treatment of physical, mathematical, numerical and application aspects. It is assumed that the student possesses undergraduate-level knowledge of

- physics in general and aerodynamics in particular

- methods of mathematical analysis and calculus, in particular linear algebra,

vector and tensor analysis, ordinary and partial differential equations - methods of numerical (mathematical) analysis

- informatics.

The contents of the course is of course limited in width and depth, in the sense

that the emphasis is on methods for stationnary subsonic and transonic flows.

The student is encouraged to consult the following literature for further study.

General References

0.1 Moran, J. 'An introduction to theoretical and computational

aerodynamics'

John Wiley & Sons,

₁₉₈₄

ISBN O-471-87491-4

0.2 Katz, J.

'Low Speed Aerodynamics, from Wing Theory to

Plotkin, Panel Methods'

Mc Graw-Hill, Inc., ₁₉₉₁

0.3 Hirsch, C.

0.4

Hirsch, C.

0.5

Hoffmann, K.

0-2

'Numerical Computation of Internal and External

Flows'

Vol. 1, Fundamentals of Numerical Discretization

John Wiley & Sons,

1988

ISBN

_{0-471-91762-1}

'Numerical Computation of Internal and External

Flows'

Vol.

2,

Computational Methods for Inviscid and

Viscous Flows

John Wiley & Sons,

1990

ISBN

0-471-92351-6

'Computational Fluid Dynamics for Engineers'

Engineering Education System, Austin, Texas,

1989

ISBN

09623731_i4l

0.6

Nixon, D. (editor) 'Transonic Aerodynamics'

Progress in Astronautics and Aeronautics, Vol. 81

AIAA, Wash,

1982

ISBN

_{0-915928-65-5}

0.7

Habashi, W.G. (editor) 'Advances in Computational Transonics'

Recent Advances in Numerical Methods in Fluids,

Voi. 4;

Pineridge Press, Swansea, UK;

₁₉₈₅

ISBN

0-906674-28-X

0.8

Henne, P.A. (editor) 'Applied Computational Aerodynamics' Progress in Astronautics and Aeronautics,

Vol. 125

AIAA, Wash,

1990

1. INTRODUCTION AND OVERVIEW

1.1. Numerical Aerodynamics; what, why and how?

Every student that is confronted with a new subject will probably immediately

feel the following questions rising within himself: What is this about?

Why is it there? How is it done?

To answer these questions completely probably takes a life-time but at least a

full course of lectures.

About 'WHAT?'

Numerical (or computational) Aerodynamics is a sub-discipline of Computational Fluid Dynamics ('CFD' for short).

CFD is concerned with research, development and application of methods for constructing numerical approximations to solutions ('numerical solutions') of

the partial differential equations that describe the motion of fluids and gasses

within or around bodies.

Numerical (aircraft) aerodynamics is concerned with the application of CFD in

aircraft aerodynamics.

Numerical Aerodynamics is also

- sitting at a graphics work-station to

generate and inspect geometries

generate and inspect meshes and grids (which form the basis of numerical

discretization) see figs. 1.1 - 1.5.

'post-process' (visualize) the results

- 'digging'

in listings of

codes to find the 'bug' - sitting at your desk to

think about

how to attack a problem (make a plan!) the error you made

call the guy that gave you the wrong data

call the guy that developed the code because it would not run (properly)

l-2

Numerical Aerodynamics, as a separate sub-discipline, is generally recognized to

have started around 1970. It owes its existence in the first place to the spectacular developments in computer technology. Processing speed and memory

increase by a factor 10 in roughly every seven years (fig. 1.9, ref. 1.1).

About 'WHY?'

Numerical Aerodynamics, as compared to windtunnel testing offers complementary

possibilities that. may improve the efficiency of the aerodynamic design process.

In particular it can lead to

- reduction of time and/or cost of design and development

- product/quality improvement

- improved accuracy of performance estimates in the early stages of the design

process (aircraft manufacturers must give performance guarantees to airlines

that 'buy from the drawing board'; hence reduced risk).

COMPETITIVE AIRCRAFT DESIGN IS NO LONGER POSSIBLE WITHOUT NUMERICAL AERO!

(everybody does it!).

About 'HOW?'

In the process of development and application of numerical methods for solving physical problems we may distinguish the following steps

definition and description of the physical problem

formulation of the corresponding well-posed analytical-mathematical

problem (mathematical-physical model)

formulation of a numerical model that discretises (approximates)

the

analytical-mathematical model and its solution which sufficient accuracy

and efficiency

formulation of algorithm(s) (set of calculation rules) for the efficient execution of the solution process for the numerical model

y) coding of the algorithm(s)

('writing' the computer program (or 'code'))

verification (checking that the code is mathematically and numerically

sound and self-consistent)

validation (checking and establishing the boundaries

of the area of

1.2.1 Sketch of Physical Problem

In numerical (aircraft) aerodynamics with emphasis on stationnary subsonic and transonic flows we focus in particular on the following physical problem (fig.

1.10):

The (air)flow about an airplane configuration (or part of a configuration)

exhibiting, in general - lift and drag

- friction effects that are limited to a relative thin layer (boundary layer)

adjacent to the surface of the configuration (high Reynolds numbers)

- 'smooth' flow 'separation' at the (sharp) trailing edge of wing-like

components and a thin 'trailing vortex wake' - compressibility effects

- disturbances that, at least in subsonic flow (Mach number <1,) vanish at _a

large distance from the configuration (except, possibly in the _{vicinity of the} trailing vortex wake).

Some of phenomena sketched above (thin boundary layers, separation at trailing edge) are typical for attached flows. However, since _{the performance envelope of}

an aircraft is usually determined by flow separation upstream of the _trailing

calibration (tuning of free parameters in the code to improve the

efficiency and/or applicability) application of the code

The mathematical-physical model plus the numerical model and algorithm(s)

constitute what is generally called a 'flow solver'.

In case (candidate) codes are already available steps iii) to vi) or vii) can be replaced by 'selection of code'.

In the following paragraphs we will look at some of these steps in a little more

detail.

edge we are also interested in the (much more complicated) problem of separated

flows. A particular kind of (controlled) separated flow characterised by the presence of strong leading-edge vortices

is typical for certain types of

military (fighter) aircraft.

1.2.2 Mathematical-Physical Models

Formulation of the mathematical(-physical) model implies

- choice of a system of conservation laws of physics, or Partial Differential

Equations (PDE's) derived there from, such that the physical phenomena that are considered to be essential are modelled.

- formulation of boundary conditions and additional conditions;

such that a well-posed mathematical problem is obtained with a unique and

physically relevant solution.

The conservation laws or PDE's model the conservation of the flow quantities

mass

momentum ₍₃ components!)

energy

supplemented with the

thermal and caloric equations of state (usually for a perfect gas)

and, for viscous, heat conducting fluids, expressions for the stress tensor

viscosity coefficient

heat conduction coefficient

Conservation laws for flow quantities U within a volume with bounding surface S can in the absence of internal and boundary surface sources, be written in the integral form

f UdQ + d = O

Q S

or in the differential form

In eqs. (1.1.1), (1.1.2) represents the amount of quantity crossing the

boundary surface. That is called the flux. In case U is a scalar quantity, such as mass or energy, the flux f' is a vector. If U is a vector quantity, such as

momentum, the flux is a tensor.

Distinction can be made between fluxes due to transport of fluid (convective fluxes) and flux due to molecular motion and thermal agitation (diffusive fluxes). The diffusive flux can also be considered as the remaining flux when the fluid is at rest; it appears only in the energy equation. Another type of

distinction is between 'inviscid' and 'viscous' fluxes. The viscous fluxes are due, directly, to viscosity and heat conduction. The inviscid fluxes are those

present in the absence of viscosity and heat conduction.

Convective fluxes contain both inviscid and viscous parts, diffusive fluxes a

'viscous' (heat conduction) part only. The inviscid parts of convective fluxes are sometimes referred to as advective fluxes.

In fluid mechanics systems of conservation laws or equivalent PDE's are usually

named after the form of the momentum equations (equations of motion) that is

utilized (e.q. Euler eqs., Navier-Stokes eqs.).

A hierarchy of mathematical-physical models cari be distinguished (fig. 1.11) in which at lower 'sub-ordinate' levels the equations are simplified successively.

The process of simplification implies a loss of physically significant

information. This has two important consequences Certain physical phenomena are no longer modelled

In a mathematical sense the problem may no longer have a unique solution. If the problem does not have a unique solution it is generally not suitable for

numerical treatment (the algorithm may not be able to choose between multiple

solutions). Additional information will then have to be supplied in order to be able to obtain a solution at all (preferably the physically relevant solution _in which we are interested).

Overview of' mathematical-physical models

The most complete description of the flow of a fluid continuum is given by the

Time-dependent Navier-Stokes equations. The table below lists the phenomena

Model eq(s) Phenomena that are modelled

Time-dependent pressure, inertia and friction forces Navier-Stokes convection (advection), diffusion,

(N-S) dissipation, heat conduction

rotation (vorticity)

(Direct simulation) separation (vortex formation)

compressibility (shock waves)

transition turbulence

The time-dependent Navier-Stokes equations can be written in the form (see eq.

(1.1.2))

au

-+ v.( + ) = o

at

mv

visc

1-6

or, expressed in Cartesian coordinates, as

au a

+ - (F. + F . )

+ -

(G. + G . ) + p (ri. + H . J = O (1.1»)

at ax

mv

visc ay

mv

visc az iriv visc

Here U represents the (5x1) column vector of conserved

quantities (mass,

momentum (3x), energy) and ' a generalized (5x3) flux vector.

F, G and H

represent the Cartesian components of F.

In a body-fitted coordinate system , n, Ç they can be written as

au ₊ a

- (.

+ )

+ - (.

+ J

+ -

(iTi. + H ) = 0 (1.1.5)

at a

mv

visc an

mv

visc aç

mv

visc

with U = TJ/J

=

k1

+ G +

G = (n F + n G + n H)/J (1.1.6)

= + G +

where J = a(,n,ç)/a(x,y,z) is the Jacobian (matrix) of the transformation from Cartesian to body-fitted coordinates.

It is noted that the energy equation for a viscous, heat conducting fluid

expresses that the local rate of change of total energy (kinetic plus internal)

is the result of the sum of transport of total energy through convection, transport of heat through conduction and the mechanical energy delivered by

viscous forces.

Furthermore, the more detailed structure of the equations implies (ref. 1.2) that the transformation of kinetic energy into heat through viscous forces

(dissipation) is irreversible. The same holds for the transformation of heat

into kinetic energy through heat conduction. As a consequence of these

irreversible processes the entropy of a (moving) particle of fluid cannot

decrease (2nd law of thermodynamics).

Numerical flow simulation based on the time-dependent N-S eqs. is often referred to as direct simulation.

The practical and biggest problem of direct simulation is formed by the fact

that almost all flows of aerospace interest are turbulent. Turbulent flows are

characterized by the presence of time and space dependent fluctuations of all

the flow quantities. In many cases the level of' the turbulent fluctuations can attain 10% or more of the mean values of the flow quantities. The biggest part

of the problem is, however, in the fact that space- and time-scales of the turbulent fluctuations can vary enormously. The numerical resolution, with

sufficient accuracy, of the small-scale fluctuations in particular is a

formidable computational task. It requires such small space- and time steps that for all practical purposes the computational effort is prohibatively large for a

long time to come, even for the biggest and fastest supercomputers.

Reynolds -averaging

The next (lower) level in the hierarchy of flow models therefore involves

application of a time averaging process to the turbulent fluctuations in order

to obtain laws for 'mean', averaged, turbulent quantities. The

time-averaging is to be done in such a way that the time-dependence of other

1-8

For compressible flows density-weighted averages of quantities A are introduced

by setting (1.1.7) T/2 A = f A(x,y,z;t+T)d-t (1.1.8) -T/2 and A = A + A" (1.1.9) pA" = 0 (1.1.10)

Hence, the time-average of the fluctuating part is set to zero.

Application of this time-averaging process to the (full) Navier-Stokes equations

leads to the Reynolds-Averaged Navier-Stokes equations (RANS).

The RANS eqs contain all the terms of the original time-dependent N-S eqs

applied to the mean flow plus a number of additional terms. The additional terms arise as a result of the non-linear character of the N-S equations. They appear where (vector) products of quantities are to be taken, as a consequence of the fact that the average of a product of fluctuating quantities,

pA"xA" 0 (1.1.11)

even if

pA" = O

(The average of a product is not the saine as the product of averages).

In the momentum equations the additional terms are interpreted as

Reynolds-stresses (turbulent Reynolds-stresses). Reynolds Reynolds-stresses appear in the equations in a

p

way similar to the viscous stresses. Hence, the RANS eqs can be written in a form similar to eq. (1.1.4), (1.1.5). In the energy equation additional terms

appear that are related to (turbulent) heat conduction.

Unfortunately the HANS eqs no longer form a closed system of equations; the

number of unknowns is no longer equal to the number of equations because of the

appearance of additional terms of the type (1.1.11). Apparently, the

time-averaging process gives rise to 'loss of information'. This loss of information

must be compensated for by explicitly adding external information from other

(experimental) sources.

The 'art' of turbulence modelling is now to 'close' the system of equations by

relating the additional, turbulent terms (Reynolds stresses) in some way to the mean flow quantities.

At this point it can be mentioned that there is an intermediate form of

numerical flow simulation based on the (RA)NS equations that is known as Large Eddy Simulation (LES). In LES, the time-step and grid-scales _{are chosen such} that the large-scale turbulent phenomena are resolved implicitly by direct simulation, while sub-grid scale turbulent phenomena are treated through a

turbulence model as in RANS.

The table below lists the flow phenomena that are modelled through the RANS

1-lo

Model eq(s) Phenomena that are modelled

Time-dependent As N-S,

Navier-Stokes with but only for phenomena with length and time sub-grid scale scales of the order of computational steps space/time averaging (grid) and larger

(tLarge Eddy Simulation, sub-grid scale phenomena through explicitly

LES) added turbulence model

Reynolds-Averaged As N-S,

(Time-averaged) but only for phenomena with time scales much

Navier-Stokes larger than time scales of turbulence phenomena

(RANS) transition and turbulence completely through

explicitly added models

It must also be mentioned that there exist two (simplified) subsets of the RANS equations that are often used in (the emerging) practice. One involves a thin

shear layer approximation leading to the Thin layer N-S equations (TLNS). In TLNS it is assumed that the dominating influence of the viscous and turbulent terms come from the gradients transverse to the main flow direction, which would be appropriate for thin shear layers. In terms of eq.(1.1.6) the TLNS eqs take the form aF. aG. au

mv

p- (H + FI . _} = O

-+

+ + at a aç

mv

visc (1.1.12)

where Ç is in the body-normal direction.

The use of TLNS is justified when the streamwise viscous terms are small and/or

cannot be resolved on the computational grid. The penalty accompanying TLNS is

They offer a reduction of computational effort in case of supersonic flow with small cross-flow and no streamwise flow separation (slender configurations). In

such conditions the character of the equations is predominantly parabolic, which opens the way for an efficient, marching-type of solution procedure.

Zonal Modelling

applies only to steady flow. The PNS equations can be written in the general

form 3F. 3G.

mv

mv

3 + + an

inv1i

.

) =0

visc (1.1.13)

It appears that further simplifications of the (RA)NS equations _{are not possible}

without distinguishing and separating viscous and inviscid parts in the flow field. It was first recognized by Prandtl

(1904)

that at high Reynolds numbers

without significant flow separation the (direct) influence of the viscous and

turbulent shear stresses is limited to a thin layer close to the wall (boundary layer) and that outside these layers the flow behaves _{as inviscid. In Prandtl's} theory a simplified boundary layer approximation (of the NS equations) suffices

for the determination of the viscous effects, while the (indirect) effects thereof on the outer inviscid flow can be represented through the concept of (boundary layer) displacement thickness. More recently Prandtl's theory has been reformulated in terms of the theory of matched asymptotic expansions (see, _e.g.

ref.l.3).

Clearly such zonal modelling requires some form of interaction between the

boundary layer computations and the computation of the outer inviscid _{flow; the} 'inner' and 'outer' solutions must be matched at a common interface: the edge of the boundary layer.

Prandtl's theory also teaches that for attached flow with thin boundary _{layers a}

fully inviscid approximation is a valid and consistent

_{one for many flow}

INVISCID FLOW MODELS

The Euler equations

The most complete inviscid flow model is obtained by setting the viscous and

heat conduction terms in the NS equations equal to zero, which gives (see eqs. (1.1.4), (1.1.5)) or au

mv

aG.

mv

8H.

mv

-+

+ +

-O

at ax ay az au ₊ a

(.

) +

- (,

+ !_ (T ) = at a

mv

an

mv

az

The table below lists the phenomena that are modelled.

Model eqs Phenomena that are modelled

1-12

(1. 1 .14)

(1.1.15)

It is important to note that by neglecting viscosity and heat conduction we have

lost the 2nd law of thermodynamics and the implicit modelling of separation

(vortex formation), as well as other viscous effects.

Hence we must anticipate that we may have to reintroduce such lost information in some other, explicit manner to obtain a problem with a unique and physically

relevant solution. An example is the condition of Kutta-Youkowski which states that the flow must 'separate smoothly' at the (sharp) trailing-edge of airfoils

and wings.

Euler eqs pressure and inertia forces

convection (advecticn)

rotation (vorticity)

The Full Potential Equation

The next lower level of approximation for inviscid _{flow is obtained by assuming}

irrotationality, leading to the Full Potential (FP) _equation:

+

.(pU) = o

p = p(U)

_(1.1.16)

U =

J

The most important aspect of the full _{potential equation is that it contains}

only one dependent variable: the velocity potential . Recalling that the Euler

equations form a system of 5 equations with _{5 dependent variables this is,} indeed, a considerable simplification.

_{The price to be paid is a further}

reduction of the number of phenomena that _{are (properly) modelled (see below).}

Model eqs _{Phenomena that are modelled}

Linearised Potential equations and Laplace's equation

If, at the bottom of the hierarchy,

_{in addition to the assumption}

of irrotationality the flow field is assumed _{to consist only of a weakly perturbed} uniform flow we obtain the Linearized Potential or Prandtl-Glauert equation

Full Potential Eq as Euler eqs but

without rotation (vorticity)

shock waves only accurate as long as they are weak

For M= O eq.(1.1.17) as well as (1.1.16) reduces to Laplace's equation + +

=0

xx yy zz = i -=

- U.x

(1.1.17)

Ihi « ü

This is a linear partial differential equation which offers a further

computational advantage as compared to the FP equation.

Model eqs Phenomena that are modelled

linearized as Full Potential Eq but with

Potential Eq compressibility effects modelled only in as

('Praridtl-Glauert Eq) far as they are linear (no shock waves)

,

+$

+4

=0

xx yy zz

or (1.1.18)

4 + +

=0

xx yy zz

Hence Laplace's equation is valid for incompressible flow, irrespective of the

level of flow perturbation.

Boundary Layer equations

As mentioned earlier the direct effects of viscous _{and turbulent stresses at}

high Reynolds numbers are confined to

_{a thin layer adjacent to the wall}

(boundary layer). In such situation the velocity _{component normal to the body}

can be argued to be much smaller than the components paralel to the wall. Introducing this assumption into the TLNS equations (1.1.12) _{leads to the}

conclusion that the equation for the normal component of momentum

reduces to

Model eqs _{Phenomena that are modelled}

Laplace Eq _{as Full Potential Eq but}

without compressibility effects

a a a a 2

(pw) +

_{(puw) + - (puw) + - pw}

_{+ p + H}

. ) = O

aç _visc (1.1.19)

where PeRfl) represents the

pressure at the edge between boundary layer arid

outer inviscid flow.

The implication of (1.1.20) is that the

_{pressure no longer appears as a}

dependent variable but as a known external driving force which is to be obtained from an inviscid flow computation.

The set of equations obtained in this _{way is known as the Boundary Layer (BL)}

Equations. In the symbolic notation of _{eq.(1.1.5), (1.1.6) they take the}

saine

form as the TLNS equations (1.1.12).

_{However, the vector U of conserved}

quantities and generalised flux vector components contain one unknown less (the pressure p) and less terms.

aç

(1. 1 . 20)

l-16

It is further noted that as with the N-S equations we may distinguish between

Time-dependent, LES and Reynolds Averaged Boundary Layer equations.

Model eqs Phenomena that are modelled

Time-Dependent as N-S,

BL Eqs but only on scale of b.l.

(i.e., e.g., no shockwaves) and only in as far

as compatible with pressure/velocity field of' external inviscid flow

LES BL Eqs as N-S/LES

idem (as Time Dependent BL Eqs)

A further simplification of the boundary layer equations is obtained if certain assumptions are made with respect to the velocity profile in the boundary layer.

In that case the BL Eqs may be integrated beforehand in the normal direction across the boundary layer, leading to the Integral Boundary Layer Equations

Model eqs Phenomena that are modelled

Integral BL Eqs as Re-Ave BL Eqs

but only for selected class of velocity profiles Re-Averaged as N-S/Reynolds Averaged

Interaction between boundary layer flow and external inviscid flow

Solving the boundary layer equations is opportune only in conjunction _with solving the equations for the outer inviscid flow. In doing so the boundary

layer solution and outer inviscid flow solution should interact and _{match at the} edge of the boundary layer.

Matching requires that in the boundary layer computations _{separation takes place}

at the same location as assumed for the outer inviscid flow _{computation. Broadly}

speaking this requirement is more or less automatically _{met in the case of}

attached flows about airfoils and wings in which both the boundary layer flow and outer inviscid flow separate at the sharp trailing edge. (In the outer inviscid flow 'separation' is effectuated through the 'Kutta _{condition'). In}

this situation one speaks of 'weak interaction' between boundary layer and outer inviscid flow.

In case, in the boundary layer computations, separation _{takes place upstream (or}

downstream) of the point where it was assumed to

_{take place in the outer}

inviscid flow one speaks of 'strong interaction'. In this _{case viscous (finite}

Reynolds number) effects have a large impact _{on the circulation and the outer}

inviscid flow (with Kutta condition) no longer represents

_{a valid first}

approximation.

In the latter situation the boundary layer computations _{can no longer be excuted} in the classical way with prescribed _{pressure. However, it appears that they can}

still be performed if the pressure is 'relaxed' in such _{a way that it can}

'adjust' to the location of the separation point. (For details see part II.)

App1icabi1it

The various flow models mentioned above all have their _{own area of applicability}

in aircraft aerodynamics. Qualitatively this _{can be indicated as in fig. 1.12.}

The figure illustrates the areas in the a-Mach plane within the flight envelope of a transport-type aircraft where the various flow models are valid.

The Navier-Stokes equations are valid in the whole of the a-Mach or CL_Mach plane (provided the representation of turbulence _{is adequate). The other flow}

models have more limited regions of applicability _{in the sense that the higher}

the model ranks in the hierarchy of fig. 1.11 _{the larger is its region of}

1.2.3. Numerical Models

In order to be able to 'numerically solve' a (system of) PDE('s) for given

boundary conditions we must first DISCRETIZE the problem. That is, the number of

unknowns (in principle infinitely large if the solution is to be known in any

point of the domain in which the solution is sought) must be reduced to a finite

number.

There exist several alternative ways of discretization, each with its own

advantages and drawbacks.

One popular class of discretization methods proceeds as follows

Define values of the function(s) that we wish to determine (e.g. velocity,

potential) on a regular spatial mesh or grid (fig. 1.13).

Approximate the derivatives of function(s) in the grid points by means of

finite differences

The requirements that the differential (or rather difference) equation(s) must

be satisfied in the grid points and that the boundary conditions must be

satisfied in the grid points on the boundary leads to a large (1O to 106)

system of generally non-linear algebraic equations. The matrices corresponding with (linearisations of) this system of equations are sparse and exhibit a band

structure. Special solution techniques exist that take advantage of this

structure.

Methods of this type are called Finite Difference Methods (FDM).

i-18

Instead of the differential form we can also apply the integral form of the

conservation laws directly to a volume element or cell of the space in which the

solution is sought.

For example, in case of the mass conservation law, this leads to (fig. 1.114)

ff1 div p Ç'cIQ = 1f p '.n dS = O

c e

Q S

e e

in which we recognize Gauss' divergence theorem.

The surface integral in (1.2.1) can be approximated numerically. For example by defining values of the (as yet unknown) flux vector pV at suitably chosen 'nodal points' in the computational grid and expressing the variation of the integrand over the cell faces by means of a Taylor series expansion around the nodes. Satisfying eq (1.2.1) for all volume elements, _{while satisfying the boundary} conditions for the fluxes in the cell faces at the boundary, one obtains, again, a system of algebraic equations with a bandstructured _matrix.

Methods of this kind are called FINITE VOLUME _{METHODS(FVM).}

The variation of the integrand along the _{cell faces can also be expressed}

in

terms of (suitably chosen) polynomials with the function values in the nodes as

parameters. This case is, sometimes, associated with the notion of FINITE

ELEMENT METHODS IFEMJ. However,

_{the notion of finite element methods is}

generally restricted to a class of methods

_{that is based on so-called}

variational principles.

As an example consider the Laplace equation

div. grad = O

(1.2.2)

It can be shown that solving (1.2.2) is _{equivalent with minimizing the volume}

integral

f!! [grad ]z dQ

(1.2.3) Q

The volume integral (1.2.3) _{can be approximated numerically by dividing} Q in

volume elements and choosing suitable (local) _{polynomials for}

with the nodal values of _{as (unknown) parameters. Minimization of the}

discretized form of (1.2.3) leads again to a system of _{algebraic equations with a band structured}

matrix.

FEM's of the type just sketched have their origin in structural analysis. _They are not very popular in numerical aerodynamics.

The main advantage of FEM's is that, _{unlike FDM's, they do not}

require regular or structured grids. This means that _{they are attractive for application}

to

irregular geometries (as often _{occur in structures). Another and}

l-20

advantage is that the manual as well as the computational effort required for

the generation of the unstructured spatial grids is significantly less than for regular, structured grids. A disadvantage is that the structure of the resulting

matrices is also less regular than in case of FDM's. As a consequence the

computational effort associated with FEM's is generally larger.

FVM's can also be formulated such that they can make use of unstructured grids. Such 'unstructured-grid' FVM's have gained in importance since the late

1980's.

For a more general introductory discussion on the differences and similarities between FDM, FVM and FEM see e.g., ref. 1.2.

Certain PDE's, for which elementary solutions are known, can be transformed into

an integral equation (by means of Green's theorem, about which later, in section

2.1.6). The solution of a boundary value problem for such PDE's can then be expressed in terms of integrals over boundary and volume distributions of elementary solutions (sources/sinks, vortices) of, as yet, unknown strength; e.g.

= JI oCx.) K(x.,x.)dS + III o(x ) K(x.,x.)dQ

(1.2.4)

13

S

13

k

where o(x.) represents the elementary solution in a point x.

and K(x.x) an

influence or distribution function associated with o(x.).

For homogeneous linear PDE's, such as Laplace's equation, the volume integral disappears, so that a distribution over the bounding surfaces suffices.

The boundary integral can be discretized on a surface mesh; for example as in a

FEM (fig. 1.15).

Requiring that a boundary condition for is satisfied in each element of the

surface mesh leads to a system of (10 to 10') linear algebraic equations for

the unknown source/sink or vortex strengths. In this case the associated matrix

is full ( no zero entries).

Methods of this kind are called BOUNDARY ELEMENT or PANEL METHODS. In Numerical (Aircraft) Aerodynamics they play an important role since about

1970.

1.2.4 Relative computational efforts

Computational aerodynamics methods based _{on the (Reynolds-averaged)} Navier-Stokes equations, the Euler equations and the Full Potential equation _are

usually of the finite difference or finite volume _{type. Those based on Laplace's} equation are of the 'Panel' type.

Numerical algorithms for the various classes _{of methods, in particular those for} the Euler and Navier-Stokes equations _{are (still) being improved continuously.}

Fig. 1.16 provides a crude impression, in _{a relative sense, of the levels of}

computational effort required for the _{different classes of methods. The figure} illustrates that the computational effort _{associated with panel methods and}

RANS-methods differ by several orders of _magnitude.

1.3

Aspects of application

Computational aerodynamics methods _{can also be classified according to their}

role in the aerodynamic design process. On may distinguish: DIRECT (or ANALYSIS)METHODS

developed for the purpose of computing the _{flow around a configuration of} given geometry (digital electronic windtunnel).

INVERSE METHODS

developed for the purpose of computing _{the (detailed) geometry required to}

generate a given pressure distribution (design). (AERODYNAMIC) OPTIMIZATION METHODS

('flow' solver plus numerical _{optimization algorithm)}

developed for the purpose of computing _{the (detailed) geometry required to}

obtain given aerodynamic characteristics _{(e.g. minimum drag).}

For obvious reasons, cat. _{(1) methods are also called}

ANALYSIS methods while cat. (2) and

₍₃₎

are called DESIGN methods. Clearly, the complementary

pos-sibilities of CFD (relative to those of windtunnel testing) are most pronounced for cat (2) and

₍₃₎

_{methods (Ref. 1.3).}

l-22

The COMPUTER POWER required by the various classes of methods varies strongly

and depends on

- mathematical/physical model (type of PDE's)

- numerical model

- required numerical accuracy of solution - turn-around time required

Fig. 1.17 gives an indication of the computer time required by various models

for one flow computation for a wing-fuselage configuration. A modest

'engineering' level of accuracy has been assumed and several types of existing

computers are considered.

Fig. 1.18 illustrates the computer capacity, in terms of processing speed and

memory required for a 'turn-around' time of about halve an hour, again assuming

a modest engineering accuracy.

Apart from a 'flow solver' the application of CFD requires extensive facilities

for PRE- and POSTPROCESSING such as

- GEOMETRY HANDLING - GRID GENERATION

- GRAPHICS for VISUALIZATION of input (geometry, grids) and output (numerical flow visualization)

The complexity of the complete CFD process requires an INFORMATION SYSTEMS

approach involving METHOD BASE and DATA BASE MANAGEMENT, EXECUTIVE and GRAPHICS

SOFTWARE, etc (fig. 1.19).

The COSTS associated with CFD development and application are substantial.

For instance, the investment associated with the development of a professional, production oriented computer program is of the order of one to several million

US $.

The direct computation cost involved with a single flow computation (3D) may

1 .4 State-of-the-art

The current

(1992)

status of numerical aerodynamics (in the Netherlands in

particular) can be summarized as follows. Routine applications (in industry) of - Panel methods for subsonic flow (including complex 3D configurations)

- 'Full Potential' FD/FV methods for transonic flow (2D and simple 3D

configurations (wing-body))

- Boundary layer methods; on a basis of, both, 'weak' (2D and 3D) and 'strong' (2D) interaction.

- 'Euler' FV methods for 2D and 3D flows with rotation. In stage of development

- 3D boundary layer methods on a basis of strong interaction - Reynolds-averaged Navier-Stokes methods.

A substantial portion of the current development effort is devoted to improved, more efficient grid generation techniques.

FUTURE PROSPECTS

Because of continuing developments in computer technology, informatics and numerical mathematics, Numerical Aerodynamics still has an enormous growth potential.

Rapidly increasing possibilities for the numerical simulation of complex viscous flows stress the need for improved turbulence models.

The need for both 'simple' (that is fast and cheap) and accurate (that is

computationally intensive and expensive) methods will remain, due to different

requirements with respect to processing speed and accuracy in the various _stages of the aerodynamic design process.

With the objective of' improving design integration computational _aerodynamics

methods will, increasingly, be integrated with computational models from other

References

l-24

AIAA J. Vol. 17, Dec. 1979, pp. 1293-1313.

1.1 Holst, T.L. The NASA Computational Aerosciences Program -Salas, M.D. Toward Teraf lop Computing AIAA-92-0558, Jan. 1992. Claus, R.W.

1.2 Hirsch, C. Numerical computation of Internal and External Flows, Vol. 1. John Wiley & Sons, 1988 (ISBN 0 47191762 1).

1.3 Van Dyke, M.D. Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, 1975.

1.4 Slooff, J.W.

Windtunnel tests and aerodynamic computations,

thoughts on this use in aerodynamic design

AGARD-CP-210, paper 11, 1976.

2. PANEL METHODS

2.1 Mathematical foundations

Panel methods are numerical methods for the solution of partial differential

equations governing potential flow, i.e. inviscid, irrotational flow. They are based on surface distributions of sources and vortices or doublets. Within the restriction of potential flows such methods can be applied to compute the flow

past complex geometries in two and three dimensions. These geometries are

approximated by a large number of surface elements or 'panels'. Depending on the number of panels any degree of accuracy may be obtained in principle.

2.1.1 Basic equations

We restrict ourselves to steady incompressible flow and to steady compressible

(subsonic and supersonic) flow subject to small perturbations.

Steady, incompressible potential flow is governed by Laplace's equation,

+ +

=0

xx yy zz

where is the total velocity potential defined by the velocity vector V;

V = grad (2.1.2)

and the subscripts denote double differentiation with respect to x,y or z, the coordinates of a right handed coordinate system. For steady compressible flow the linearized potential equation may be written in terms of the perturbation

potential ' as

(2.1.1)

(1 - M2) + + = o

xx yy zz (2.1.3)

where

is defined by the free stream velocity U _{and the total velocity}

potential as

2-2

and M is the free stream Mach number. The linearized potential equation is also called the Prandtl-Glauert equation. The potential equations (2.1.1) and (2.1.3)

follow from the law of mass conservation

div (pg) = 0 (2.1.5)

through the assumption of irrotationality. The latter is valid for homentropic

flow and is expressed by eq. (2.1.2). For incompressible flow in which the

density p is constant we find

div grad = 0 (2.1.6)

which is the same as Laplace's equation, eq.

(2.1.1). In terms of the

perturbation potential given in eq. (2.i.14) we obtain

+ + = 0 (2.1.7)

xx yy zz

If the potential ' or is known, the pressure may be obtained from the momentum equation. For incompressible flow this takes the form of the Bernoulli equation

1 _{+ ipU2}

p +

p

(2 +

_{x y} +

2) =

_{z 2}

PP,

or, in terms of the pressure coefficient C

-pU2 + 2 ₊ 2

-1 = 1

IvIz

p U2 U_ 1

_(U+

)2 + + =

= [i

_M2 [ X y z U2

(2.1.8)

(2.1.9)

In compressible flow with small perturbations where the perturbation velocity components are written as

u,v,w = , , << U

(2.1.10)

X y ¿

the density in isentropic flow is given by (ref. 2.1)

or from -P

_1pU

2 4) p

_y

= p U [-y + 4) z p

= pU [+

Z U_o

Substituting these expressions into eq. (2.1.5) we obtain

. [(i-w)

4) ]

+ .

[4) j + [ J

= o

ax

x ay 3z 1

1M2

P0

2°

the components of p'? in the continuity equation (2.1.5) are then expressed in a

series expansion of 4) 4) , 4) as X

y

z 4) = p U [i + (1-M2) ₊ U o (U + 4) )2 + 4)2 + 4)2 o

= (i

L Y z]T-1 Po

_iM2 I

O (2.1.12) (2.1.13)

which is the conservation form; in non-conservation form the Prandti-Giauert equation, eq. (2.1.3) is obtained. Eq. (2.1.3) is valid for subsonic (M<1) as well as supersonic (M>1) flow. For the present we consider only subsonic flow. The pressure is obtained from eq. (2.1.11) as

(2.1.14)

(2.1.15)

Note: In the remaining of this chapter we will understand by and 4) the

Expanding eq. (2.1.14) in a power series

P=1_Mz (2+ ..)

p

2 U

we obtain the approximation for the pressure coeficient,

C = - 2 - +

p U

Since Laplace's equation is the limit for M40 of the Prandtl-Glauert equation,

eq. (2.1.3), the latter has a larger range of applicability than the first.

Laplace's equation is valid for incompressible, irrotational flow, whereas the

Prandtl-Glauert equation may be applied in compressible, irrotational flow

subject to small perturbations and in incompressible, irrotational flow with no restrictions regarding the perturbations. Therefore most panel methods are based

on the Prandtl-Glauert equation. In general the pressure is computed fom eqs.

(2.1.14) and (2.1.15), which reduce to the exact expressions

(2.1.8) and

(2.1.9), respectively, for M+O. Note that this feature is not present

if

instead of eqs. (2.1.14) and (2.1.15), eqs. (2.1.16) and (2.1.17) are used.

The Prandtl-Glauert equation (2.1.3) may be transformed into Laplace's equation

by the transformation X' =

= y

(2.1.19) = z or by V V + V V +

xx

yy

zz'

=0

2-4 (2.1.16) (2.1.17) (2.1.18) X' = X = (2.1.20) =

where = 1_M2.

In the following we therefore confine ourselves first to the treatment of panel

methods for Laplace's equation; the description to be given may be transferred directly to the Prandtl-Glauert equation through the (inverse) transformation. We will return to the subject of compressibility effects in section 2.3.3.

Both Laplace's and the Prandtl-Glauert equation equations are linear partial differential equations, which allow superposition of solutions to create new solutions. The latter property is utilized extensively in potential flow theory and as a consequence also in panel methods.

2.1.2 Boundary conditions and additional conditions; uniqueness of the _boundary value problem*)

Types of problems

In order to obtain a specific solution for the potential _{of Laplace's equation}

one has to impose boundary conditions. The latter have to be chosen in such a way that the solution represents the desired flow situation (in our case the flow past an aircraft or part of it).

For Laplace's equation two types of problems may be distinguished in terms of

topology:

- problem for an interior domain (internal flow), fig. 2.1.1 - problem for an exterior domain (external flow), fig. 2.1.2 Also two types of boundary conditions exist:

- Neumann-type boundary conditions, expressing that the normal derivative of the

potential on the boundary, (which is equal to the normal velocity _component

(=.n)), is prescribed, fig.

_{2.1.3. A problem involving this type of}

boundary conditions only is called a Neumann-problem. The _{tangential velocity} component at the boundary and thus the pressure distribution may be calculated when the solution of the boundary value problem is known. The Neumann-problem

arises if one wishes to compute ('analyse') the flow about a body of given

geometry. It is therefore called an analysis problem

- Dirichiet-type of boundary conditions, expressing that the potential on the

boundary is prescribed, fig. 2.l.k. Prescribing the potential

implies

prescribing the tangential velocity component by the derivatives and

where s,t are coordinates along the boundary. In this case the normal velocity

component is obtained from the solution. The Dirichlet problem is related to

the problem of designing the shape of a body for a given velocity or pressure

distribution. Well-posed problems

1f

dS = O an s 2-6

A condition for (numerically) solving a boundary value problem is that the

problem is well-posed or properly-posed (ref. 2.3), by which we understand that

a continuous and twice differentiable (C2) solution exists;

at any point in the domain where that solution is sought it is unique;

there exist neighbouring solutions, i.e. for slightly different boundary

conditions the solution changes only slightly.

The properties with respect to well-posedness of some typical Dirichlet- and

Neumann problems are summarized below. For a proof one is referred to text books

on partial differential equations or to ref. 2.2.

Well-posed is a

- Dirichlet-problem for an interior domain Q bounded by a closed surface S (not intersecting itself), fig. 2.1.5.

- Dirichlet-problem for an exterior domain, fig. 2.1.6. This problem may be

obtained from the problem shown in fig. 2.l.6b by letting

S*.

Not well-posed is a

- Neumann-problem for an interior domain bounded by a closed surface, fig. 2.l.7a. It can be shown that there exists a solution only if the additional

condition

(2.1.21)

is satisfied,

indicating

that inside the closed surface S no mass is created or destroyed.

The solution for the potential that exists inside S under the condition (2.1.21) is not unique, because an arbitrary constant, _c say, satisfying

Laplace's equation, may be superimposed without violating the boundary

condition = O (grad = O).

an c

- Neumann-problem for an exterior domain. This problem may be obtained from that

depicted in fig. 2.l.7b for

S9.

_{The not well-posedness of the}

Neumann-problem for an exterior domain with = O on S. (fig. 2.1.7b) and S* seems,

at first sight, to constitute a severe handicap for applications to aircraft

aerodynamics where we are particularly interested in such problems. However, we shall see shortly that this is not really the case.

The mixed or Poincar problem for an interior domain bounded by a closed surface

(fig. 2.1.8) may be shown to be well-posed. No additional condition for

_{j needs}

to be prescribed. The mass flow imposed on a part of the boundary by prescribing

is absorbed by the part where

is prescribed. The latter guarantees

uniqueness of the potential and as a consequence the uniqueness of the problem.

Note that in order to satisfy Laplace's equation must be twice continuously differentiable in points of the boundary (A and B in fig. 2.1.8) where the type

of the boundary conditions changes. Because of the relevance for the aerodynamic computations encountered in the aeronautical design problems we will analyse the mixed boundary value problem in some more detail.

Consider the earlier mentioned case of a multiply connected domain shown in figs. 2.l.5b and 2.1.7b, but now as a mixed boundary value problem with _a Neumann boundary condition on S _{and a Dirichiet condition on S0, see fig.} 2.1.9. Independent of the choice of on S, any solution satisfies the Neumann condition (for example = O) on S..

i

The question now is how to select the potential on S in order to have the desired model that simulates the flow around the closed surface S., i.e. how _we

have to select

= g(s,t) on S where g(s,t) is a known function of the

coordinates s,t on S. To answer this question we consider the physics of the flow past an aircraft configuration (fig. 1.10). From that _{we learn that at}

large distances from S. the flow conditions approach those of the _undisturbed

flow. In other words the normalized potential and perturbation _{potential at} S = S behave like

or = g(s,t) 4 X B !! _{ds =} A as 2-8 for S (S ) (2.1.22) o

Whether this condition holds everywhere on S remains to be analysed. For

example nothing has been said about the trailing vortex wake of the

configuration.

The mixed boundary value problem with a Neumann boundary condition at the body

surface 5B _{= Si} and a Dirichlet condition expressing zero perturbation potential at infinity appears to be the appropriate problem to solve for our applications. However, there is at least one important shortcoming, as we will discuss in the

next section.

2.1.3 Modeling of circulation/lift

In order to illustrate the shortcoming of the flow model mentioned in the

previous section we introduce a plane Q intersecting SB and S, fig. 2.1.10;

the intersections are aS and aS, respectively. We determine the line integral along the curve as from A to B (fig. 2.1.10) as

(2.1.23)

The double brackets ] denote a jump in the value of the quantity inside. Realizing that a potential flow does not contain rotation, eq. (2.1.23) may be recognized as the expression for the circulation r around a cross-section 35B of

5B (see text books on elementary aerodynamics).

For a continuous potential along as (as well as 3SB) we have

r = hm

0 (2.1.24)

B9A

In other words there is no circulation and therefore no lift.

The only way to introduce circulation is to allow for a discontinuity in the

potential between A and B. However, this cannot be done without due

considera-tion, since it would be incompatible with the requirement

that (s,t) is

continuous on 5. This inconvenience is circumvented by introducing a cut asc

(4

between the points A and B on as and the cross-section aSB of the body. Across the cut the potential is allowed to jump from a level to

2' according to

= h(s,t) (on Sc) _(2.1.25)

where we have formally assumed that the jump is a function h(s,t) of the local coordinates. In fig. 2.1.11 a sketch is given of the model representation with _a cut. The circulation around the sectìon

35B of' fig. 2.1.11 is then

(F)as

L

(2.1.26)

Note that the introduction of the cut has reduced the original

multiply-connected domain into a simply-multiply-connected domain. Without proof _{we mention here}

that a well-posed problem with circulation can only be formulated for _a

simply-connected domain. Every cross-section of the type eQ needs a cut to produce a

well-posed problem with circulation; in three dimensions we thus obtain a

discontinuity surface Sc extending from the body SB to the surface at 'infinity'

S, see fig. 2.1.12. The discontinuity surface intersects the surface S _{at a} line s _{(fig. 2.1.12). In general it will not be possible for both the upper and} lower side of S

_{to blend smoothly with the body SB. thus violating the}

requirement that the bounding surface of Q must not intersect itself for a

properly posed boundary value problem (section 2.1.2). As _{a consequence we may} expect that special measures must be taken in order to cope with this _situation, about which later.

We may further notice, that since (s,t) should be continuous over the bounding surfaces, we must require, at the edges of the discontinuity surface _{Sc, that}

ir = o _(2.1.27)

In addition it may be clear that because of the jump in _{the potential across Sc} we have to abandon the Dirichlet condition (2.1.22) on S _{in the vicinity of s;} we come back on this later.

and a a

[p + (-)

]j

= o

a a

LE íp) j:B1

=

o

a 3 2 LE

(p)

j Ill

0 2-10

Boundary conditions on the discontinuity surface

The requirement to have a jump in the potential across S is insufficient to determine the solution of the mixed boundary value problem uniquely. For a

complete description, either 4 should be prescribed on both sides of Sc, or the

normal derivative on Sc, should be given, or an equivalent of both. How can

we achieve this?

The physics of the flow require conservation of mass, momentum and energy. This should also hold across S.

Conservation of mass across SC involves that no mass is created nor destroyed,

which can be stated as

= o

or, in case of incompressible flow

= ° (2. 1 .28)

(normal velocity component is continuous across Sc).

It can be shown further (see, e.g., chapter

3)

that in an irrotational flow which is uniform at infinity upstreai conservation of energy is satisfied

implicitly when conservation of mass is satisfied. Hence, it remains to consider

conservation of momentum.

Conservation of momentum across S may be formulated as

(tangential momentum)

(normal momentum) (2.2.29a)

(2. 2. 29b)

(2.2.29c)

Using (2.1.28) this can be reduced to

and a a P

= o

a a p

j

= o

respectively. Since a a a a =

and because [] is not necessarily constant on S (if it were, it would have to be zero, because of (2.1.27), it follows that (2.2.30 b,c) _{can be satisfied only}

if both

(p Jl = _(2.2.31a)

and

an2

= (2 .2. 31b)

We note that eqs (2.2.31 a,b) express that S

_{must be a stream surface.}

Unfortunately we do not know the exact shape and position of a stream surface a priori, without knowing the solution. What _{we do know, however, is that if the} flow separates somewhere on the body a stream surface eminates from the line at which the flow separates into the flow field. In case of sharp-edged bodies, like a wing the flow is known to separate _{at the sharp (trailing) edge. This} then suggest that a surface approximating the _{stream surface from the sharp} (trailing) edge represents a proper _{a priori choice for the discontinuity}

surface S. In case of small perturbations _{streamlines in the flow will almost} be parallel to the undisturbed flow. In such conditions S may chosen to consist

of straight generators, parallel to the x-axis, eminating from the trailing edge.

(2.2.30b)

+ 2 + _{= O}

IL-n s t 1 on Sc

2-12

With such an a priori chosen shape and position of S the conditions (2.2.30a)

and (2.2.31a and b) are one too many for a properly

posed boundary value

problem. In addition we have the complication that the zero pressure jump

condition (2.2.30a) is a non-linear one. The general approach is then to replace

the two conditions (2.2.31 a plus b) by the single (weaker) condition (2.1.28). This, of course, still guarantees conservation of mass.

Using the Bernoulli eq. (2.1.8) the non-linear zero pressure jump condition

(3.2.30a) can be written as

onS0

(2.1.32)

(which is the saine as (

(II)2

= O or O.

For small perturbations (see eqs. (2.1.16), (2.1.17)

this reduces to the

approximation

cP = O on S

(2.1.33)

which is equivalent with

= const. in x-direction on S (2.1.34)

The nonlinear character of condition (2.1.32), complicates matters considerably, therefore, in practice, linear approximations are used of which eq. (2.1.33) or eq. (2.1.34) is an example.

An alternative may be found in

writing eq. (2.1.32) in terms of 'normal'

and tangential coordinates as

(2.1.35)

These expressions may be used if we posess a more accurate estimate of the flow

direction on Sc than the free stream direction. or, using eq. (2.1.28),

lt + = O

OflSc

The coordinates (s,t) may then be chosen such that

f

= 0 In this case the s- direction, denoted by s, coincides with that of the component, in S, of the

average streamline of S (see fig. 2.1.13). Eq. (2.1.36) then reduces to the linear expression

or

* it:

Ji

= const along s

The boundary conditions on S are summarized in fig. 2.1.14.

It is finally noted that matters simplify considerably in the case of

two-dimensional flow. In that case tIe one remaining tangential momentum condition,

as well as the normal momentum (zero pressure jump) condition, are automatically satisfied when = 0; this irrespective of the shape arid position of S.

Kutta condition

In the preceding section we have shown that a jump in the potential across the discontinuity plane Sc is necessary in order to model circulation. However, the magnitude of the jump has not yet been stipulated. It appears to be sufficient

that the jump [ ] is given along SB _{the intersection of 5c with SB (see fig.} 2.1.12); the distribution on Sc then follows from eqs. (2.1.34) or eq. (2.1.38).

considering again the physics of our problem (fig. 1.10) we observe that the

condition that the flow separates 'smoothly' from the (sharp) downstream end _SB

of' the body has not been satisfied yet. Here, basing on heuristic, physical arguments, 'smoothly' is to be interpreted as with finite velocity () that is

continuous when passing from the body onto S across the trailing edge.

This condition is known as the condition of Kutta-Youkowski _{or simply as the}

Kutta condition. The situation suggests that

JJ along SB be chosen such that

the Kutta-condition is satisfied.

At this point it should be noted that the more formal argument that Laplace's

equation (2.1.1) or (2.1.6) must be satisfied everywhere in the flowfield, the line

5b on the bounding surface SB+SC included, leads to the same requirement, namely that is finite and continuous when passing from _{onto Sc. Hence, we}

[L

* :11=0

on Sc s

expect that this is the governing condition for determining the circulation.

Continuity of velocity implies continuity of across the trailing edge, thus

for P1, P, P2, P T (P ) 1 (P2) - (P) + for P1, P, P2 P2 + T (2.1.40)

-Subtraction of these expressions yields

- (P1) (P) - (P) = =

vS-

(2.1.41)

or

2-14

may conclude that if we make sure that Laplace's equation is satisfied at the

trailing edge we automatically will also satisfy the Kutta condition.

Recall also in this context, that, in section 2.1.3, we anticipated that special

measures were to be taken at the intersection of S with SB in order to restore

the conditions for a properly posed boundary value problem (that is that the

solution is continuous and differentiable twice ("C2") everywhere in the flow field). Apparently, the Kutta condition serves as such.

Finite and continuous velocity requires in the first place a continuous

potential across the trailing edge, which can be expressed as (see fig. 2.1.15

for the position of the points P)

and or (Pj) . for P1, P, P2, P -* T and - (P1)

_ii:

for P1, P, P2, P + T (2.l.39a) + (P1) (P) + (2.1 . 39b)

and

where the latter equality results from the condition that _ff = O across Sc;V denotes the gradient in S. Eq. (2.1.41) puts a requirement on the jump of

the tangential velocity across S.

Addition of the expressions (2.1.40) results in

+ (P1) 9 (P) + (2.1.42)

which puts a requirement on the average velocity.

Multiplying the upper part of eq. (2.4.41) with +

_{(P) and the lower}

part with V(P1) + e(P) we obtain

p,

-

[(P1))2

+

(()}2

-

(P))2

=

2 ₂

On the surface of discontinuity we have, according to eq.

(2.1.35),

2j = O.

Hence, it follows that

((P2))2

-

[(P1)J2

O (2.1.43)

The condition given by eq. (2.1.43) may also be obtained by requiring continuity

of pressure, because this means (according to the Bernoulli equation, eq.

(2.1.9)), see fig. 2.1.14 ( + + 2 )

= (2 +

Z )

(2 +

+ 2 )

x

y

zP2

s

tP2

s t

nP2

(

P1PjP2P2

T

(2 +

+ ) ( + z )

(2 +

+ 2 ) , (2.1.44) x y

zP

s

tP

s t

np

J

where on the body surface (p1, P2) there holds = O

Taking into account the jump condition, eq. (2.1.35) we find

(2 +

2)

(2 +

Z) p

5

tP

p'

1,P24T

(2.1.45)

which expresses the same as eq. (2.1.43). These conditions are nonlinear; taking

for s the streamline direction (or a good approximation of it), eq.

(2.1.45)

and at a at 2-16 ( s -; P1,P2 T (2. 1. 46)

which, in small perturbation theory, may be approxinated as

(o ) 4 (0 )

X Pl X P1,P2 T

Eqs. (2.1.143, 146, 47) are sometimes referred to as 'equal pressure' forms of the

Kutta condition. Note that they (pre-)assume that (2.1.39) is already satisfied.

In order to be able to appreciate the implications of (2.1.41) in some further

detail we will consider the individual components

of in 'normal' and

'tangential' directions n, s and t respectively, where we choose the s direction

normal (Sn) and t parallel to the trailing edge. Introducing unit vectors n,

and respectively eqs. (2.1.41) may then be written as (see fig. 2.1.15)

a (Pa) caS 02 as (P2) sin 2 ' (Pi) n a a j-;: (P2) sin 02 _as (P2) cas 02 (P n a

9(P2)

j

cas 01 - ---- (P1) sin _fl (Pi)

n

9

---- (P1) sin 0

as

(P1) cos 01

as

(Pi)

n n

Because we have -- = O at P1, P2, eqs. (2.1.148/49) may be reduced to

(2.1.47)

(2.1.48)

Because the cut surface must be a streamsurface (or at least a good

approximation to it)

(p1, P2) must also be zero. It then follows from

(2.1.50a) and (2.l.51a) that

both

as

(P2) sin 82 0 ' for p1, p2 T (2.1.52) and --;; (P1) sin 0 O J

This can be satisfied only under either one of the following three

conditions

-ç (P2) sin 2 (Pa) 17 for Pl, P2 T (2.l.53a) ::n (P1) O 8s (P) n (2.1.50) (b) ----

(P2) cos 02

9-n (P

9(P2)

at 2 J (c) and

4--an

(Pi)

(a)

as

(P1) sin 01

n (2.1.51) (b) --- (P1) cos O 9 1 n

as

(Pi)

n at ' 1 -: (E'1) -J (c) (a)