Michele and Alberto FANELLI Inviscid model of a plane jet...OBTAINED BY A CONFORMAL MAPPING (A2) ......

Michele & Alberto FANELLI

MATHEMATICAL SIMULATION OF A PLANE JET THROUGH FUNCTIONS OF A COMPLEX VARIABLE

An example of the representative power of this schematization: - Synthetic overall features of the motion field - Immediate analytical & graphical representation





SECTIONS OF THE PRESENTATION:

A - AN EXAMPLE OF APPLICATION OF THE PROPERTIES Of FUNCTIONS OF A COMPLEX VARIABLE :

THE PROBLEM OF THE PLANE JET

B - ESSENTIAL BASES OF AN INTRIGUING ISOMORPHISM :

FUNCTIONS OF A COMPLEX VARIABLE PLANE IRROTATIONAL MOTIONS OF AN INVISCID, INCOMPRESSIBLE FLUID

C – ADVANTAGES, LIMITS & DRAWBACKS OF THIS TYPE OF SCHEMATIZATION





WHY USE THE FUNCTIONS OF A COMPLEX VARIABLE TO APPROACH THE

STUDY OF SOME PROBLEMS OF PLANE HYDRODYNAMICS ?

Before any theoretical background, let us begin directly by a provocative example:

A - AN EXAMPLE OF USE OF THE FUNCTIONS OF

A COMPLEX VARIABLE:

THE PROBLEM OF THE PLANE JET

AN EXAMPLE : APPLICATION OF FUNCTIONS OF A COMPLEX VARIABLE TO A PROBLEM OF PLANE HYDRODYNAMICS : THE DIFFUSION OF A PLANE JET

ISSUED FORTH INTO AN INVISCID LIQUID BODY (A1)

THE PROBLEM – Many situations of interest concern the steady immission of a liquid discharge from a point-like outlet into an indefinitely extended liquid body.

Variables to be investigated are , e. g., the angle of diffusion of the jet, the flow perturbations in the receiving pond, the mixing of the jet with the surrounding liquid environment …In current practice use is made of empirical-statistical correlations , or of a priori assumptions … but what about a detailed mathematical description of the flow field ?....

THE PHYSICS – Turbulence promotes mixing in the ‘far field’, while in the ‘near field’ the individual large eddies tend to maintain their distinct identities.

WHAT POSSIBLE MATHEMATICAL IDEALIZATION?

THE IDEA OF A POSSIBLE INVISCID MODEL BASED ON A CAREFULLY CHOSEN FUNCTION OF A COMPLEX VARIABLE :

….A CONFORMAL MAPPING OF THE VON KARMAN «VORTEX STREET» ….. ….Plausible for the ‘near field’? Let us try and see!...

THE PROBLEM OF THE PLANE JET : AN INVISCID, IRROTATIONAL APPROXIMATION (A2)

STARTING POINT : VON KARMAN «VORTEX STREET» (from PRANDTL):

THE PROBLEM OF THE PLANE JET, FIRST STEP: VON KARMAN «VORTEX STREET»

OBTAINED BY A CONFORMAL MAPPING (A2)

Only 4 eddies

An

infi

nit

y o

f e

dd

ies

Plane

Plane

Conformal mapping

λ.π = ln 1 + 2 ≅ 0.8813736…

THE PROBLEM OF THE PLANE JET . VON KARMAN «VORTEX STREET» : GRAPHICAL REPRESENTATION OF THE FLOW FIELD (A3)

- - - - - = equipotentials 𝜑 = 𝑐𝑜𝑛𝑠𝑡. ; = streamlines ψ = const.

Function 𝐹 ζ = 𝜑 + 𝑖. 𝜓

The flow field moves downwards with uniform

speed Г

2.𝜋. 2

Clockwise eddies on the left side,

circulation −Г

Anti- clockwise eddies on the right side, circulation + Г

λ.π = ln 1 + 2 = 0.8813736…

𝐹 𝜁 = 𝑖.Г

2 𝜆 −

1

𝜋, 𝑙𝑛

𝑆ℎ 𝜁 − 𝜆. 𝜋

𝐶ℎ 𝜁

is the function of 𝜁 = 𝜉 + 𝑖. 𝜂 and 𝜑 = 𝑅𝑒 𝐹 , ψ = Im 𝐹

VON KARMAN «VORTEX STREET» : GRAPHICAL REPRESENTATION OF THE FLOW FIELD (A4)

THE TRAJECTORIES

Microscopic indication of retrograde motion

Retrograde motions off the Vortex Street

Alignment of clockwise eddies Alignment of anti.clockwise

eddies

General motion of the Vortex Street

THE PROBLEM OF A PLANE JET : THE FINAL CONFORMAL MAPPING OF THE VON KARMAN «VORTEX STREET» (A5)

𝜑 = velocity potential, 𝜓 = stream function, both real functions of ξ and η The function representing the Von Karman vortex street"on the plane ζ ≔ ξ + 𝑖. η 𝑖𝑠 ∶

𝐹 𝜁 = 𝑖.Γ

2. 𝜆 −

1

𝜋. 𝑙𝑛

𝑆ℎ 𝜁 − 𝜆. 𝜋

𝐶ℎ𝜁= 𝜑 + 𝑖. 𝜓

A final conformal mapping brings the plane 𝜁= ξ + i.η onto the plane 𝑍 = 𝑋 + 𝑖. 𝑌 , where the

jet is to be represented , by another logarithmic transform:

𝜁 = −𝑖. 𝑙𝑛𝑍 +𝜆. 𝜋

2

i. e. 𝑍 = 𝑒𝑥𝑝 𝑖. 𝜁 −𝜆.𝜋

2 , from which is obtained the velocity potential 𝜑 of the jet on the

plane 𝑍 = 𝑋 + 𝑖. 𝑌:

𝐹 𝑍 = 𝑖.Γ

2. 𝜋. ln

𝑆ℎ𝜆. 𝜋2

+ 𝑖. ln 𝑍

𝐶ℎ𝜆. 𝜋2

− 𝑖. ln 𝑍+

Γ

2. 𝜋. 𝑖. 𝜆. 𝜋 = 𝜑 + 𝑖. 𝜓

𝜑 = 𝜑 𝑋, 𝑌 = velocity potential of the jet, from which: 𝑉 = 𝑔𝑟𝑎𝑑 𝜑

GEOMETRICAL MEANING OF THE CONFORMAL MAPPING USED TO TRANSFORM THE «VORTEX STREET» (A6)

A onformal mapping of a plane 𝜉 , 𝜂 onto another 𝑥 , 𝑦 establishes a point-to-point, line-to-line correspondence. The angles and the

singularities are conserved, not so the areas.

THE PROBLEM OF THE PLANE JET : GRAPHICAL REPRESENTATIONS OF THE FLOW FIELD OBTAINED FROM THE «VORTEX STREET» AFTER THE FINAL

CONFORMAL MAPPING: EQUIPOTENTIAL LINES (A7)

THE PROBLEM OF THE PLANE JET : GRAPHICAL REPRESENTATIONS OF THE FLOW FIELD OBTAINED FROM THE «VORTEX STREET» AFTER THE

FINAL CONFORMAL MAPPING: STREAMLINES (A8)

An

gle

of

dif

fusi

on

of

the

jet

2.𝛼

=2.a

rctan

λ.𝜋 2

= 4

7°5

64

91

…

Point of immission

THE PROBLEM OF THE PLANE JET : GRAPHICAL REPRESENTATIONS OF THE FLOW FIELD OBTAINED FROM THE «VORTEX STREET» AFTER THE FINAL

CONFORMAL MAPPING: EQUIPOTENTIALS AND STREAMLINES (A9)

Local velocity vectors

Some of the equipotentials are missing in this zone

MAIN GEOMETRICAL AND KINEMATICAL PARAMETERS OF THE JET

AS DERIVED FROM THE INVISCID MATHEMATICAL MODEL (A10)

Angle of diffusion of the jet : 2.𝛼 = 2. arctanλ.𝜋

2= 47°56491

Velocity of an eddy distant 𝜌 from the point of immission : 𝑉 ∗ 𝜌 =Г

2.𝜋.

1

𝜌. 2

«Time of flight» of an eddy from the point of immission to the distance 𝜌 : 𝑇 𝜌 =2.𝜋

Г. 𝜌2

Distance covered by an eddy from its emission to time 𝑇 : 𝜌 𝑇 =Г.𝑇

𝜋. 2

Polar distance between an eddy at 𝜌 units of length from the immission point and the nearest

preceding eddy of opposite sign : 𝐷 = 𝜌. 1 − exp −𝜋

2 (*)

Ratio between two successive such polar distances 𝐷𝑛

𝐷𝑛−1= exp

𝜋

2= 4.8104774…

Discharge of the jet over a layer of unit thickness (normally from the jet plane) :

𝑞 =Г

2.𝜋. 2..sin(2.𝛼) = 0.083059. Г

-------------------------------------------

(*) Near the immission point there is an infinity of eddies of alternate sign

THE PROBLEM OF THE PLANE JET : ITS TIME-DEPENDENCE IN THE FRAME OF

THE CONFORMAL MAPPING 𝜁 𝑍 (A11)

On the starting plane 𝜁 the Von Kàrmàn «vortex street» the flow field

(equipotentials + streamlines) is displaced downwards with uniform vertical

velocity 𝑉 = −Г

2.𝜋. 2 . In the passage from plane 𝜁 to plane 𝑍 this time-

dependence translates into the equation :

𝐹 𝑍, 𝑡 = 𝑖.Г

2. 𝜋 . ln

𝑆ℎ𝜆. 𝜋2

+ 𝑖.Г. 𝑡

2. 𝜋. 2+ 𝑖. ln 𝑍

𝐶ℎ𝜆. 𝜋2

− 𝑖.Г. 𝑡

2. 𝜋. 2− 𝑖. ln 𝑍

+ 𝜋 − 𝑖. 𝜆. 𝜋

From this equation the velocity field is obtained as a function of the space coordinates and of the time, which allows to compute (by numerical integration) the trajectories (distinct from the streamlines).

THE PROBLEM OF THE PLANEJET : ITS TIME-DEPENDENCE IN THE FRAME OF

THE CONFORMAL MAPPING 𝜁 𝑍 : THE TRAJECTORIES (A12)

Retrograde motions

A sort of ‘shear layer’ appears at the jet boundaries

No

n-s

imm

etri

cal m

oti

on

s In

sid

e t

he

jet

A finite number of fluid particles are released at time t = 0 from equi-spaced points along a semicircle, and their trajectories are followed over a finite interval of time

THE PROBLEM OF THE PLANE JET : ITS TIME-DEPENDENCE IN THE FRAME OF

THE CONFORMAL MAPPING 𝜁 𝑍 : THE TRAJECTORIES (A13)

2 4 6 8

4

2

2

4

A certain number of fluid particles positioned at time 0 along a semi-circumference is followed during a finite interval of time t > 0 by tracing their trajectories. The non-symmetric quality of the motion field, as well as the unbalancing influence of the eddies successively issued from the immission point, are clearly shown. Important retrograde motions are induced outside the jet.

THE PROBLEM OF THE PLANE JET : TIME-DEPENDENCE . THE «SMOKELINES» – A

VIRTUAL «WIND TUNNEL» ? (A14)

If the velocity field (Eulerian formulation) is time--dependent the trajectories are distinct from the streamlines

[because 𝑣 = 𝑣 (𝑋, 𝑌, 𝑡)] . They have to be computed by integration, incorporating the circumstance that

while the particle we are following is moving the underlying velocity field is changing all the time. Thus, in the

infinitesimal interval of time 𝑑𝑡 the average velocity is given (EULER trapezoidal rule) by :

𝑣𝑚 =𝑣 𝑋, 𝑌, 𝑡 + 𝑣 𝑋 + 𝑑𝑋, 𝑌 + 𝑑𝑌, 𝑡 + 𝑑𝑡

2

from which is to be computed the infinitesimal coordinate increment :

𝑑𝑠 = 𝑑𝑋 + 𝑖. 𝑑𝑌 = 𝑣𝑚. 𝑑𝑡

Yet different are the «smokelines» which in tests on physical models are obtained by rendering visible the

evolutions of liquid filaments issued from a fixed source immersed in the motion field and marked by a tracing

substance (e. g. smoke, whence the name) . These lines too can be computed by integration following the

deformation in time of the already developed stretch of the smokeline, and in this way we can rely on a sort of

«virtual wind tunnel». The graphic rendering is best obtained by computer-generated animations .

THE PROBLEM OF THE PLANE JET : GRAPHICAL RENDERINGS OF ITS TIME-

DEPENDENCE . THE «SMOKELINES» (animation) (A14 bis)

DA FARE ?

THE USE OF FUNCTIONS OF A COMPLEX VARIABILE AS A TOOL TO INVESTIGATE SOME PROBLEMS OF PLANE

HYDRODYNAMICS

B – ESSENTIAL BASES OF THE AMAZING ISOMORPHISM BETWEEN THE ANALYTICAL FUNCTIONS OF A COMPLEX VARIABILE AND THE PLANE MOTIONS OF INVISCID, INCOMPRESSIBLE FLUIDS .

WHY THE FUNCTIONS OF A COMPLEX VARIABLE CAN BE APPLIED

TO THE PLANE HYDRODYNAMICS OF PERFECT FLUIDS ? (B1)

• Sometimes it is possible to ignore the effects of viscosity (hypothesis of inviscid fluid), of the compressibility and of the vorticity : indeed,

• A motion field developing from an initial quiescent state or from a velocity

distributione deriving from a potential ( 𝑉 = 𝑔𝑟𝑎𝑑 𝜑 ), and acted upon by mass forces deriving as well from a potential , will stay irrotational (eddies cannot be generated nor destroyed, i. e. vortex filaments present in the initial

state , if any, will last forever ) ; from 𝑉 = 𝑔𝑟𝑎𝑑 𝜑 𝑐𝑢𝑟𝑙 𝑉 = 0 ;

• If the fluid is incompressibile the dominating condition (besides 𝑉 = 𝑔𝑟𝑎𝑑 𝜑 )

is 𝑑𝑖𝑣 𝑉 = 0, i. e. 𝜕𝑢

𝜕𝑋 +

𝜕𝑣

𝜕𝑌 = 0 : creation or destruction of volumes of fluid is

not allowed, except in some inextended singularites (sources and wells). In

terms of velocity components : 𝜕𝑢

𝜕𝑋 +

𝜕𝑣

𝜕𝑌 =

𝜕2𝜑

𝜕𝑋2 +𝜕2𝜑

𝜕𝑌2 = 0 .

Here is to be found the connection with the special properties of the analytical functions of a complex variable, which as a consequence become a very powerful, synthetic tool for a prima facie analysis of hydrodynamic problems.

THE ISOMORPHISM OF PROPERTIES OF ANALYTiC AL FUNCTIONS OF A COMPLEX VARIABLE (AFCV) AND THE INDEFINITE DIFFERENTIAL EQUATIONS

OF THE PLANE HYDRODYNAMICS OF INVISCID, INCOMPRESSIBLE FLUIDS (B2))

Definitions and properties of analitical functions of a complex variable (AFCV)

1° – What is an AFCV ?

Any «familiar» function of a real variable can be the ‘model’ of an AFCV.

Example: Sh X Sh (Z) , where Z = complex variable = X + i.Y and 𝑖 = −1.

In the example Sh (Z) = Sh (X+i.Y) = Sh(X).Ch(i.Y)+ Ch(X).Sh(i.Y) =

= Sh(X).Cos(Y) + i. Ch(X).Sin(Y) = 𝜑 + 𝑖. 𝜓 .

Obviously in the example:

𝜑 = Re [Sh(Z)] = Sh(X).Cos(Y) and

𝜓 = Im [Sh(Z)] = Ch(X).Sin(Y) ; both 𝜑 and 𝜓 are real functions of X and Y .

THE ISOMORPHISM OF PROPERTIES OF ANALYTiC AL FUNCTIONS OF A COMPLEX VARIABLE (AFCV) AND THE INDEFINITE DIFFERENTIAL EQUATIONS OF THE PLANE HYDRODYNAMICS

OF INVISCID, INCOMPRESSIBLE FLUIDS (B3))

2° - Definitions and properties of the analitical functions of a complex variable (AFCV)

2.1 – Property of the «complex conjugate»:

If 𝐹(𝑍) = 𝐹(X + i.Y) = 𝜑 + 𝑖. 𝜓 , then 𝐹 𝑍 = 𝐹(X - i.Y) = 𝐹 𝑍 = 𝜑 − 𝑖. 𝜓

2.2 – Very important in the applications: 𝜕𝜑

𝜕𝑋 =

𝜕𝜓

𝜕𝑌 e

𝜕𝜑

𝜕𝑌 = −

𝜕𝜓

𝜕𝑌 (pay attention to the − sign !)

Eqs. 2.2) follow from the fact that the complex derivative 𝑑𝐹

𝑑𝑍 is independent from

the direction 𝑑𝑍 = 𝑑𝑋 + 𝑖. 𝑑𝑌 along which the differentiation is performed.

In the example 𝑑[𝑆ℎ 𝑍 ]

𝑑𝑍= 𝐶ℎ 𝑍 = 𝐶ℎ 𝑋 + 𝑖. 𝑌 = 𝐶ℎ𝑋. 𝑐𝑜𝑠𝑌 + 𝑖. 𝑆ℎ𝑋. 𝑠𝑖𝑛𝑌 ;

𝜕𝜑

𝜕𝑋= 𝑆ℎ𝑋. 𝑐𝑜𝑠𝑌 =

𝜕𝜓

𝜕𝑌 ;

𝜕𝜑

𝜕𝑌= −𝐶ℎ𝑋. 𝑠𝑖𝑛𝑌 = −

𝜕𝜓

𝜕𝑋) . Notice the − sign !

THE BASES OF THE USE OF THE FUNCTIONS OF A COMPLEX VARIABLE AS A MODEL OF THE PLANE HYDRODYNAMICS OF

PERFECT FLUIDS (B4)

Eqs. 2.2) show why it is possible to put the AFCV in relation to the plane hydrodynamics of ‘perfect fluids’. Indeed, foran arbitrarily chosen AFCV 𝐹 𝑍 = 𝜑 + 𝑖.𝜓 :

2.3 - From 𝜕𝜑

𝜕𝑋 =

𝜕𝜓

𝜕𝑌 and

𝜕𝜑

𝜕𝑌 = −

𝜕𝜓

𝜕𝑋 : 𝛻𝜑 =

𝜕2𝜑

𝜕𝑋2 +𝜕2𝜑

𝜕𝑌2 = 0

which means that for any AFCV, 𝐹 𝑧 = 𝜑 + 𝑖.𝜓 , the function 𝜑 𝑋, 𝑌 = 𝑅𝑒 𝐹 can be taken as the velocity potential of a plane, irrotational motion field of a ‘perfect fluid’.

𝛻𝜑 =𝜕2𝜑

𝜕𝑋2 +𝜕2𝜑

𝜕𝑌2 = 𝑑𝑖𝑣𝑉 = 0 if 𝑉 = 𝑔𝑟𝑎𝑑 𝜑

Components of 𝑉 ∶ 𝑢 = 𝜕𝜑

𝜕𝑋 and 𝑣 =

𝜕𝜑

𝜕𝑌 = −

𝜕𝜓

𝜕𝑋 .

THE BASES OF THE USE OF FUNCTIONS OF A COMPLEX VARIABLE AS MODELS OF THE PLANE HYDRODYNAMICS OF

MOTION FIELDS OF PERFECT FLUIDS (B5)

2.4 – The 2 components of the velocity field obtain from a single operation in the complex domain :

𝑉 = 𝑢 − 𝑖. 𝑣 =𝑑𝐹

𝑑𝑍

Attention to the − sign!

In the example 𝐹 𝑍 = 𝑆ℎ𝑍 ,

𝑑𝐹

𝑑𝑍= 𝐶ℎ𝑍 = 𝐶ℎ𝑋. 𝑐𝑜𝑠𝑌 + 𝑖. 𝑆ℎ𝑋. 𝑠𝑖𝑛𝑌,

𝑢 = 𝐶ℎ𝑋. 𝑐𝑜𝑠Y , 𝑣 = −𝑆ℎ𝑋. 𝑆𝑖𝑛𝑦 ] i. e.

𝑢 = 𝑅𝑒𝑑𝐹

𝑑𝑍 , 𝑣 = −𝐼𝑚

𝑑𝐹

𝑑𝑍

The − sign is essential for correctness of applications and graphics.

THE BASES OF THE USE OF FUNCTIONS OF A COMPLEX VARIABLE AS MODELS OF THE PLANE HYDRODYNAMICS OF

MOTION FIELDS OF PERFECT FLUIDS(B6)

Type of math. model

Ideal physical model

Velocity potential

(IRROTATIONALITY):

𝒄𝒖𝒓𝒍 𝑽 = 𝟎

Incompressible fluids:

SOLENOIDALITY

𝒅𝒊𝒗 𝑽 = 𝟎

«PERFECT» (INVISCTD) FLUIDS

INITIAL STATE : QUIESCENT OR

IRROTATIONAL MOTION

YES

-------

INCOMPRESSIBLE FLUIDS

(INFINITE CELERITY)

-------

YES

PLANE INVISCID MOTONS

SIMULATION BY FUNCTIONS

OF COMPLEX VARIABLE

𝛁𝜓 = 𝟎 𝐘𝐄𝐒 𝛁𝜑 = 𝟎

THE BASES OF THE USE OF FUNCTIONS OF A COMPLEX VARIABLE AS MODELS

OF THE PLANE HYDRODYNAMICS OF MOTION FIELDS OF PERFECT FLUIDS (B7)

Equipotential lines , streamlines.

The meaning of 𝜑 = 𝑅𝑒 𝐹 𝑧 has been defined . But what is the meaning of 𝜓 = 𝐼𝑚 𝐹 𝑧 ?

The lines 𝜑 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 are equipotential lines, everywhere normal to the local velocity vectors.

The lines 𝜓 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 are streamlines , everywhere tangent to the local velocity vectors.

In the streamtube between tw𝑜 streamlines 𝜓 = 𝜓1 and 𝜓 = 𝜓2 there flows a discharge 𝜓1 − 𝜓2 for each unit of thickness normally to plane 𝑧 .

Equipotential lines and streamlines form two families of mutually orthogonal curves. (for a graphical example, see e. g. next slide B8). For equal increments ∆𝜑 and ∆𝜓 the two families draw a tessellation of plane 𝑧 with «small squares» of varying size (see again next slide).

THE BASES OF THE USE OF FUNCTIONS OF A COMPLEX VARIABLE AS MODELS OF THE PLANE HYDRODYNAMICS OF MOTION FIELDS OF PERFECT FLUIDS :

EXAMPLES (B8)

Dipole (source and well of infinite intensity infinitely near)

𝐹 𝑧 = 1

𝑧=

𝑥−𝑖.𝑦

𝑥2+𝑦2


CONFORMAL MAPPINGS (B9))

The plane hydrodynamics of a perfect incompressible fluid can be treated fully well using adequately chosen AFCV .

Features: Every motion field described by an AFCV is completely determined by its singolarities and its boundary conditions.

A single AFCV can generate a una multitude of motion fields, by means of suitably chosen «Conformal Mappings» (CM) . i. e.:

from the complex plane z = x + i.y F (z) potential 𝜑 motion field;

CM: mapping of the plane z= 𝑥 + 𝑖. 𝑦 onto the plane ζ = ξ + i.η

If the CM tra is defined by an AFCV : 𝑧 = 𝑀 ζ , the function 𝐹 𝑀 ζ = 𝑓 ζ defines the transform of the motion field on the plane ζ . There is a point-to-point, line-to-line correspondence between the two planes …..


CONFORMAL MAPPINGS (B10)

Example of a conformal mapping (CM):

Plane z : 𝐹 𝑧 = 𝑆ℎ 𝒛 𝜑 + 𝑖. 𝜓 = 𝑆ℎ 𝑥 . 𝑐𝑜𝑠 𝑦 + 𝑖. 𝐶ℎ 𝑥 . 𝑠𝑖𝑛 𝑦

Conformal mapping (CM): 𝒛 = 𝑙𝑛ζ ossia ζ = 𝑒𝑧

Put ζ in the form ζ = 𝜚. 𝑒𝑖.𝜃 (polar coordinates) ; then on

Plane 𝜻 𝑓 ζ = 𝑆ℎ 𝑙𝑛𝜻 = 𝑆ℎ 𝑙𝑛 𝝔. 𝒆𝒊.𝜽 = 𝑆ℎ 𝑙𝑛𝜚 + 𝑖. 𝜃 da cui

𝑓 ζ = 𝑆ℎ 𝑙𝑛𝜚 .𝑐𝑜𝑠 𝜃 + 𝑖. 𝐶ℎ 𝑙𝑛𝜚 . 𝑠𝑖𝑛 𝜃

On the plane ζ = 𝜚. 𝑒𝑖.𝜃 the velocity potential is therefore: 𝜑 = 𝑆ℎ 𝑙𝑛𝜚 .𝑐𝑜𝑠 𝜃


LIMITS OF THEIR REPRESENTATIVITY OF REAL SITUATIONS (B11) But what can be said about the representativity of the ideal motion fields thus derived with respect to the motions of real fluids? There is more in it than could be thought at first sight; indeed,

the motion fields derived from any given AFCV : - Satisfy the EULER motion equations. - Satisfy the principle of «least kinetic energy». - They even satisfy the NAVIER-STOKES equations for a Newtonian fluid in all those cases in which the viscosity stress components be everywhere locally equilibrated….

The main drawback is the impossibility of complying with one of the two boundary conditions at boundaries between the fluid and solid boundaries (null tangential component of velocity). However, in many cases this condition is confibed in a thin «limit layer» while the motion field ‘at large’ is well enough approximated by a potential field.


HYDRODYNAMICS

C – ADVANTAGES AND DRAWBACKS OF THIS TYPE OF MATHEMATICAL IDEALIZATION


HYDRODYNAMICS(C(C1)C1)1)

Deep roots of the representativity of AFCV with respect to the physical reality of the hydrodynamics of «perfect fluids» :

- If the viscosity effects can be disregarded, the motion field is dominated by the condition of solenoidality 𝑑𝑖𝑣𝑉 = 0 and by the boundaries geometry.

- The dynamics of the motion field is then a strict consequence of its kinematics, rather than its first cause … «geometrization of plane hidrodynamics».

- Any AFCV , because of the very nature and rules of the complex analysis, hans a real component whose gradient complies with the solenoidality condition. Compliance with the boundaryconditions can be obtained by a multitude of techniques, among which the use of CM .

The existence of a velocity potential allows to operate formally on a single scalar function rather than on a two-components vectorial field (see e.g. the use of this approach in the formalization of the dynamic fluid- structure interaction).

Drawback: the sign change in the vertical component of the velocity…


HYDRODYNAMICS(: ADVANTAGES AND DRAWBACKSC(C2)C1)1

The existence of a velocity potential allows to operate formally on a single scalar function rather than on a two-components vectorial field (see e.g. the use of this approach in the formalization of the dynamic fluid- structure interaction). In actual applications, serious difficulties can be met whenever the geometry of the impervious solid boundaries is complicated. However, there are general methods (see e. g. the Schwarz-Christoffel CM, or the «source method») which in principle allow to cope numerically with arbitrarily shaped boundaries.

«Well-practised» domains of use of AFCV :

- Thermal problems - Electric and magnetic problems - Seepage motions (loose materials dams…) - Hydrodynamics of the fluid/structure interaction - Naval hydrodynamics and subsonic Aerodynamics - Simplified internal hydrodynamics of hydraulic machinery - Problemi of plane elasticity and plane fracture mechanics …


HYDRODYNAMICS : HOW REALISTIC IN THE CASE OF THE PLANE JET ? (C3)C1)1

However suggestive, the mathematical model of a plane jet presented as an example is doubtlessly an incomplete and deformed image of the real kinematics and dynamics of real jets, insofar as: - It represents a regime situation, avoiding to treat the initial transient ( apparently paradoxical

aspects of the time-dependence ) - It is doubtlessly inadequate to represent the «far field» dominated by mixing mechanisms - The diffusion angle of the model jet is probably too wide . This last fault could be overcome by using a slightly different CM , e. g. :

𝑭 𝒛, 𝒕 = 𝒊. 𝒍𝒏𝑺𝒉

𝜦𝟐+ 𝟐. 𝒊. 𝐥𝐧 𝒛

𝑪𝒉𝜦𝟐− 𝟐. 𝒊. 𝒍𝒏 𝒛

− 𝝅 + 𝒊. 𝜦

However, the model here discussed can be used as a «first approximation» which is already interesting and anyway much more «realistic» than the elementary model of a point-like (whose diffusion angle would be 360°) ; taking this first model as a starting point ever more sophisticated models could be evolved ….

But chiefly , this example clearly shows the remarkable power and synthetic qualities of the use of AFCV, in synergy with the advanced CFD numerical methodologies .

Michele and Alberto FANELLI Inviscid model of a plane jet...OBTAINED BY A CONFORMAL MAPPING (A2) ......

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