Some Quality Information from Similitude Functional ...

IJAMP, Vol. 4, No. 2, July-December 2012, pp. 165-173

© Global Research Publications, India

Some Quality Information from Similitude Functional Notations for an External Compressible Flux Field Past a Random Shaped Rigid Body, via a Multi – Valued Calculus Formalistic Approach J. Venetis* & E. Sideridis**

ABSTRACT In this paper we will try to deduce some quality information about a generic form of external compressible flux fields, past a random shaped rigid body, about the relationship between the governing influential factors, which actually are Reynolds and Mach dimensionless numbers and the geometrical features of the rigid body, via an assessment of the related similitude functional notations. In particular, the method that we will develop here is motivated by a multi – valued Calculus approach. In particular, provided a specific rate of the ratio:

CL CD

�

���� FL ���� FD

, we will deduce some constraints for the geometrical

properties of the rigid body, which is interpolated at this flow pattern, by utilizing these two known from literature functional notations, which include the afore – mentioned governing influential factors of these flux fields. Keywords: Similitude, Functional notation, Rigid body, Drag force, Uplift force, Lateral force.

1.

INTRODUCTION

Wind is a randomly non deterministic dynamic loading. The flow of the air induces a velocity pressure of any structures, wich is taken into consideration during conceptual design procedure. Dynamic effects of natural wind are mainly introduced by a variance of wind speed and wind pressure within time and space. The complex action of wind is a combination of mutually influencing effects such as [20]: •

Wind climate (the global wind)

•

Topology and roughness of terrain (the local wind)

•

Aerodynamics, aero elasticity (drag coefficients and wind force)

•

Structural dynamics (response of the structure)

•

Design of the structure

In most compressible air dynamic flow patterns, it is accurate to assume the flux field as frictionless irrotational and isentropic throughout. Nevertheless in supersonic flows due to shock waves occurrence, isentropic condition is abolished. For elliptic boundary value problems which concern sub – sonic flow patterns, shock waves motivate singularities hence any change in boundary conditions affects the whole region of flow. * **

School of Civ. Eng. NTUAthens, E-mail: [email protected] School of Applied Mathematics and Physical Sciences NTUAthens, Section of Mechanics, 5 Heroes of Polytechnion Avenue GR – 157 73 Athens, Greece, E-mail: [email protected]

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J. Venetis & E. Sideridis

On the contrary, for hyperbolic boundary value problems which deal with ultra – sonic flow patterns, singularities may result if boundary conditions are prescribed on a closed boundary. Thus, the change in boundary conditions affects only a limited domain of flow [15]. A distributed wind load procedures a load intensity (force per unit of element of length), that is proportional to the sine of the angle between the element and the direction of loading. This is indeed equivalent to using a fixed load intensity that actually is measured per unit of projected element length. The fixed intensity would be based upon depth of wind speed. The projected element length is measured in a plane perpendicular to the direction of loading. The contribution of this paper, to the quality investigation of an external compressible flux field, past a random shaped rigid body, via the assessment of similitude functional notations, counts on some basic assumptions which are imperative, in our opinion, for the apposed mathematical formalism: ��� (i) For the Fluid: Before the leading edge of the obstacle, the free stream velocity V� is dominated, having exponential or logarithmic form withrespect to the elevation from the ground (or generally from an arbitrary datum). •

Density: � � ct (Compressible flux field)

•

Dynamic Viscosity: � = � (T0) = ct (generally verifies Sutherland’s semi – empirical law and occurs constant rate for isothermal flow)

•

Pressure (barometric + manometric): P � ct

(ii) For the rigid body: •

length: l (This is actually the length of the normal projection to axis x�x, of its cross – section with the family of planes parallel to axis z�z)

•

maximum thickness: d (This is actually the width of its cross – section)

•

angle of attack: a

Besides, we must denote inceptively, that we have neglected the lateral wind force, reducing henceforth the whole problem in the study of the cross – section shape of the rigid body, being evident with its intersection with a plane parallel to axis z�z. Thus, the evident functional notations for the coefficients of drag and uplift force, being motivated by the known from literature dimensional arguments, are apposed as follows [7]: CL = fL (Re, M ) CD = fD (Re, M ) where:

Re =

� � V� � l V� � � � (� � l ) � 1 � �

1 (� � l ) �1 � � = Re V�

and:

M=

(1.1)

V� V = � � c E �

E � 1 = M V�

(1.2)

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However, the constant magnitude E denotes Young modulus and

is the sonic speed.

For elastic continuum media the two quantities above are related as follows: c�

E �

(1.3)

As for the sonic speed, it is also known from elementary Thermodynamics that the following statement holds [15]:

c2 �

CP 1 � � P � � �� � CV � � �V �T

(1.4)

or in a more simplified way, the sonic speed inside air is: 1/ 2

� dP � c�� � � d� �

(1.5)

Besides, taking into account Euler’s equation for one – dimensional steady and inviscid flow it implies: VdV � �

dP dP d � d� �� � � � c2 d� � � �

(1.6)

Hence, we infer: d� dV � �M2 � V

(1.7)

We have also to remind here, that for one – dimensional compressible flow patterns of a perfect gas the rate of instantaneous velocity must satisfy the following inequality: � 2 � � � R* � T � V� � � � �1 � �

(1.8)

� 2 � � � R* � T � V� � � � �1 � �

(1.9)

Hence, it follows that:

where: ��

cP cV

R* � cP � cV du dh , cV � dT dT 2 u � ct h� 2

cP �

(1.10)

On the contrary with the above approach, if one does not use dimensional arguments and functional notations with respect to the characteristic sizes of the aforementioned flux field, can make indeed the alter – ego assumption that the continuum media which surrounds the rigid body, consists in a Tricomi gas.

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Thus, the inceptive original problem can be actually reduced to the quality or quantitative investigation of the corresponding mixed type PDEs, which concern sub – sonic and ultra – sonic flux fields past rigid bodies [10]. As a matter of fact, for this case we investigate here the unknown geometrical shape of the mounted obstacle can probably cause problems in the mathematical formulation of boundary conditions, especially if they are represented in weak form (i.e., Sommerfeld boundary conditions), since we cannot be asserted apriori if the bound of the cross – section verifies Jordan Criterion of rectilinearability. 2. QUALITY EVALUATION OF THE FLUX FIELD According to the afore – mentioned concept of a compressible and isothermal flux field past a random shaped rigid body, (mounted or moving), the coefficients of drag and uplift force, (we have neglected initially the lateral one), if we follow the classic dimensional arguments formalistic approach, can be represented as functional notations of two characteristic dimensionless quantities, which actually govern this flow pattern [14]: CL = fL (Re, M ) CD = fD (Re, M )

(2.1)

Besides regarding these coefficients, the two further relationships hold true:

CL � � CD �

��� FL � � V�2 �

A 2 (2.2)

��� FD � � V�2 �

A 2

where: A denotes either the frontal ��� or the���bottom surface of the rigid body, which accepts the «attack» of the fluid, by means of the forces: FD and FL correspondingly.. Hence, it is follows: A=b�d A=b� l

(2.3)

Thus, according to the above relationships, it can be also inferred that: ��� FL CL � ��� � f (Re, M ) CD FD

(2.4)

In the sequel, one can easily prove the existence of a multi – valued function: � : R2 � R, such that:

�y y � � 1 1 � , � � �� 1 , 2 � f (Re, M ) � � � � Re M � � V� V� �

(2.5)

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The above argument, can be also apposed in a more rigorous representation, via the following mathematical statement:

�y y � � 1 1 � � Re, M � R : M � 0.3 � � : R 2 � R : f (Re, M ) � � � , � � �� 1 , 2 � � Re M � � V� V� � � y1 , y2 � R Apparently, we have already made the following substitutions: y1 � (� � l )� 1 � � � y2 �

(2.6)

E �

Next, with the functional equation (2.5) in hand, we are able actually to be led to an equivalent PDE,���via � the following process, provided that you first let us consider as constant, or generally as datum, the ratio:

FL ���� FD

.

Thus, we can write out: � y y �� 1 , 2 � V� V�

� � � ct � �

� y y � �1 � 1 , 2 � � 0 � V� V� �

(2.7)

where: �1 consists in a binary relation in implicit form, which is actually satisfied by the real values: V�, y1, y2. Furthermore, we can remind from elementary multi – valued Calculus, that for every implicit relationship in the generic form: f (x, y, z) = 0 the necessary and sufficient condition in order this to be solved univocally with respect to the value: z, at an «adjacent» topological sphere of R3 which can be actually illustrated as a Cartesian product, (in proportion with the familiar from single – valued Calculus corresponding representation, of �-neighborhood): {(a1 – �, a1 + �) � (a2 – �, a2 + �) � (a3 – �, a3 + �), past a particular point with coordinates: (a1, a2, a3) � Df � R3 is: f ( a, b, c ) � 0 � � f ( a , b, c ) �0 �z

(28)

Hence, at the particular case we study, we can write out: y1 = y1 (V�, y2)

(2.9)

However, we have chosen as dependent variable here, the value y1 just because it contains indirectly the geometrical size: l which actually generates some first information for the shape of the rigid body. Based on eq. (2.7) and taking also into account eq. (2.9), we can deduce by means of elementary multi – valued Calculus, that the following implication holds true:

170

J. Venetis & E. Sideridis y1 � y1 (V� , y2 ) �y y � �1 � 1 , 2 � � 0 � � V� V� � �y �y y1 � 2 � V� � 2 � y2 � y1 � y1

(2.10)

In the sequel, we can recur at eq. (2.9) making the further assumption that the multi – valued function: y1 = y1 (V�, y2) is locally invertible with respect to the value: V�, in the same topological sphere that we have already supposed for the existence of the univocal solution of implicit eq. (2.7) with respect to the value y 1. Obviously, if the above assumption does not hold true, we can focus on the intersection of these regions, provided that it is not the empty set. Substantially, this latter assumption implies that: V� = V�(y1, y2).

(2.11)

The above assumption can be actually asserted, necessarily and sufficiently, via the validity of the following condition: f ( a, b, c ) � 0 � � f ( a, b, c ) �0 �V�

(2.12)

Thus, according to the above data, we can actually develop the following mathematical statement:

�V V � Re, M � R : M � 0.3 � � : R 2 � R : f (Re, M ) � f � � , � � y1 y2

� � y1 V� � � 1 � � � �� , �M � � �� , � � Re � � � V� y2 � ���� FL

Then, keeping considering as constant, or generally as datum, the ratio: ���� and repeating the same FD process as previously, we can be led afterwards to an implicit function in the particular form:

� y V � �1 � 1 , � � � 0 � V� y2 �

(2.13)

Henceforth, it is standing that: V� = V�(y1, y2). Then, by means of multi – valued Calculus, one can easily prove that the following implication also holds true: V� � V� ( y1 , y2 ) � y V � �1 � 1 , � � � 0 � � V� y2 � y1 � V� y2 �V� � � � �1 V� � y1 V� � y2

(2.14)

The simultaneous validity of these two evident relationships: (2.10) and (2.14) can actually motivate some restrictions for the geometrical features of the rigid body, which ��� is� interpolated at the investigated compressible flux field, for each circumstantial given rate of the ratio:

FL ���� FD

.

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Thus, we can utilize them in consistency to one another, (as an algebraic system), taking concurrently into account the known conditions which concern the partial derivatives of invertible multi – valued functions. Hence, we can infer: y1 �

� y2 �y � V� � 2 � y2 � y1 �V� �

y1 � V� y2 �V� � � � �1 V� � y1 V� � y2 V� �

� y2 �y � y2 � y1 � 2 �V� � y1 �

y2 � V� y �V � �1� 1 � � V� � y2 V� � y1 � �y � � y �V � � y2 � � y2 � y1 � 2 � � � 1 � 1 � � � � y1 � � V� � y1 � �

(2.15)

where, we can recall from eq. (2.6) the afore – mentioned substitutions: y1 � (� � l )� 1 � � � y2 �

E �

Next, eliminating the term of density � from the above relationships we obtain: y1 � � y2 E � l

(2.16)

Taking also into account that: E = ct and l = ct , we can eventually result to the following expression: � y2 � y2 �� �y � 1 1 � 3 � 1 � � � � 2 � � 2 � �� 2 � � 2 � y1 �� � y1 �� l y1 2 l y1

(2.16A)

Besides, because the distribution of free stream velocity: V�, (which is actually motivated by the approximate solution of Ekman’s system for the atmospheric boundary layer [18]), is assumed to be dependent only on the elevation from the ground, it is also evident that: �V� �0 �y1 ��� However, for flow patterns����� past� non planar rigid bodies, the free stream velocity: V� is replaced indeed by the external local velocity: Ve (l )����� and � Reynolds number by the corresponding local one, which depends on the algebraic rate of the vector Ve (l ) .

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Therefore, eq. (2.15) yields eventually to the following form: � y �y � � �V (l ) � y2 � � y2 � y1 � 2 � � �1 � 1 � e � � � y1 � � Ve (l ) � y1 � � � 1 �3 � 1 � � y �V (l ) � y2 � � y2 � � � 2 � � � � � 1 � 1 � e � l y1 � � Ve (l ) � y1 � 2 �

(2.17)

Hence, with the above quality information in hand, we are able to deduce some constraints for the length of the normal projection to axis Ox of rigid body’s cross – section. Besides, we can also take into account that the ratio of the geometrical attributes: dl generally underlie in the technical restrictions which are motivated by the basic principles either of airfoil engineering or architectural conceptual design, according to the occasional particular statement. Thus, we can also deduce some information about rigid body maximum thickness: d. In addition to the above arguments, we can also recall from elementary Gas Dynamics theory, that Bernoulli equation holds for compressible flows and can be exhibited for one – dimensional flow patterns in the form: d � dV ds � � �0 � V s

(2.18)

where the variable s denotes here the instantaneous perpendicular distance between the bound of the obstacle and the first upward free stream surface. This surface, actually consists in a rigid lid through which only molecular mass transfer takes place. This latter relationship, evaluated in accordance with eq. (1.7) yields: ds dV s � V 1� M2

(2.19)

Then, provided that, M > 1 implies ultra sonic speed and 0 < M < 1 implies subsonic speed, we can effectuate this latter expression in accordance with the evident eq. (2.17) in order to infer some final constraints for the designing length of the normal projection to axis Ox of rigid body cross – section. 3. DISCUSSION In this paper, we attempted to deduce some quality information for a generic external compressible flow pattern past a random shaped obstacle, about the relationship between the governing influential factors, (which actually are Reynolds and Mach dimensionless numbers) and the geometrical features of the rigid body, via the assessment of the related similitude functional notations, implementing an Applied Analysis approach. However, the convergence of stream lines in a three dimensional flow field past a rigid body with random shape, has several times significant changes in comparison to the corresponding two dimensional flow field, so the whole problem could be investigated alternatively in Cartesian space, by designing an isometric projection of the mounted obstacle and following CFD methods, provided of course that it is known either the geometry of rigid body or the instantaneous velocity distribution on the circumstantial bound of it.

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Note also, that this bound must consist in a simply closed, not self – intersecting and continuously contracted graph throughout, verifying concurrently Jordan Criterion of rectilinearability. ��� Besides, for flow patterns past non planar rigid bodies, the free stream velocity: the V � is replaced by �� ��� � external local one: Ve and Reynolds number by the corresponding local one, which depends indeed on Ve . Hence, by means of eq. (2.17) we may be able to result more information about the geometrical attributes of the occasional rigid body, provided that we have taken as datum the each time algebraic rate of the ���� quotient:

FL ���� FD

.

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