Ceva Model

INCAS BULLETIN, Volume 4, Issue 4/ 2012, pp. 85 92 ISSN 2066 8201

A New Mathematical Model for

Coand Effect Velocity Approximation

Valeriu DRGAN*

*Corresponding author

*POLITEHNICA University of Bucharest, Faculty of Aerospace Engineering Str. Gheorghe Polizu, nr. 1, sector 1, 011061, Bucharest, Romania

[email protected]

Abstract: This paper addresses the problem of obtaining a set of mathematical equations that can

accurately describe the velocity flow field near a cylindrical surface influenced by the Coand effect. The work is relevant since the current state of the art Reynolds Averaged Navier Stokes models with

curvature correction do not completely describe the properties of the flow in accordance with the

experimental data. Semi-empirical equations are therefore deduced based on experimental and

theoretical state of the art. The resulting model is validated over a wider range of geometric layouts

than any other existing semi-empirical model of its kind. The applications of this model are numerous,

from super circulation wing calculations to fluidic devices such as actuators or fluidic diodes.

Key Words: Coanda effect, fluidic device, CEVA, super circulation, regression, semi-empirical

equation, wall jet.

1. INTRODUCTION

The Coand effect is encountered in virtually all aerodynamic applications and can be described as the tendency of a fluid to attach itself to a curved wall (oriented away from the

direction of flow) Ref [1]. Some authors [2, 3] propose the calculation of the pressure drop

over the ramp trough a balance between pressure forces and centrifugal forces as shown in

Fig.1 acting upon a unit volume on its curved trajectory imposed by the wall curvature.

The aeronautical applications of the Coand effect date back to Ion Stroescus patented upper surface and trailing edge blowing high lift devices Ref [4, 5]; however, the technical

applications extend to many other fields such as fluidic and micro fluidic devices Ref [6 ,7].

Nevertheless the first attempts to describe the flow fields near a Coand effect ramp were semi-empirical Ref [8, 9, 10], lately the most prominent methods used by authors Ref [11-

14] are numerical RANS models which are increasingly easy to use both because of the

developing computing power and because of the level of sophistication offered by curvature

correction viscosity models such as the SARC Ref [15], k-omega SST RC Ref [16, 17] and

others.

However, the curvature correction in most RANS models targets the turbulence

production or (specific) dissipation terms in order to accurately estimate the separation point

of the flow which in the standard models is usually overestimated Ref [18]. These approaches do not offer a correct velocity flow field around the curved surface when

compared to the experimental data Ref [19, 20]. It is therefore the purpose of this paper to

determine a set of simple semi-empirical equations that being generated from experimental

data, can correctly match the physical measurements for wall jets subject to the Coand effect.

DOI: 10.13111/2066-8201.2012.4.4.7

Valeriu DRGAN 86

INCAS BULLETIN, Volume 4, Issue 4/ 2012

Fig. 1 The geometrical setup of a curved wall jet

2. THE PROPOSED MODEL

Essentially, most experimental studies that have been carried out through the 1980s and 90s bear the following traits: 1. The flow is thin, typically the blowing slot height is less than 6% of the curvature

radius R.

2. The atmosphere is quiescent i.e. the velocity of the ambient air is null.

3. The blowing velocities are low, usually less than 50 m/s

The proposed model, CEVA (Coand Effect Velocity Approximation), is based on the observation that thin wall jets display self-similarity as shown in Fig.2 Ref [21, 22].

This is a strong indicator that the boundary layer is laminar since self-similarity criteria

are not encountered in turbulent boundary layers due to the more complex velocity profile

development Ref [23].

As in the case of the other semi-empirical models, a radial velocity distribution is

determined, describing the normalized u/um as a function of y/y1/2.

Knowing that the velocity profile is self-similar we can describe the true velocity profile

for each individual angular location by determining the respective maximum velocity um as

well as the respective reference thickness y1/2. For achieving this, the practice is to describe

the two key parameters, um and y1/2, as a function of the h/R ratio and the circumferential

distance from the blowing slot, x-x0.

In the case of the local maximum velocity, the equation is typically expressed as a ratio

between um and the initial blowing velocity uj.

Another parameter that is explicitly calculated is the wall jet boundary layer thickness,

ym. All of the existing models regard ym as linear dependent on y1/2, which is true since we

assume the self-similarity of the profile. However, in the far out regions of the ramp for angular positions higher than 180, the boundary layer starts growing at a higher rate due to the transition from laminar to turbulent Wygnansky Ref [24]. It should be noted that in

Lewinsky and Yehs definition the ym notation represents the thickness for which the local maximum velocity is found. Later on, the notation was changed by Rodman et al. to denote

the boundary layer thickness ( therefore the velocity encountered at ym is equal to 99% of the

maximum local velocity um).

Jet

FC (Centrifugal Forces)

FP (Pressure Forces)

Plenum

Pressure

R

h uj

um()

87 A New Mathematical Model for Coand Effect Velocity Approximation


Fig. 2 The normalized radial velocity distribution as presented by Wygnansky Ref [22].

In theory, as well as in practice, the portion in the near proximity of the blowing slot

does not exhibit the self-similarity properties; this is linked with flow development and limits

the models to within a distance of at least 20 away from the blowing slot. Another exception is signaled by Wygnansky Ref [24] and refers to the transition from

laminar to turbulent boundary layer in the very far regions of the ramp, typically at angular

positions higher than 180. This does not influence the applicability of the model since most wings that use the

Coand effect both Upper Surface Blown Wings Ref [25] and entrainment wings [26] maintain the jet attached on a section ranging from 80 to 110. Early models describe the radial velocity distribution with two separate equations, for

the boundary layer and for the far field, e.g. the Rodman Wood Roberts (RWR) model:

(1)

(2)

Note that for the purposes of this paper, the boundary layer thickness is noted with ym

rather than the classical . This change in notation is generally accepted because the fluidic jet may be used in conjunction with an external flow, in which case the symbol will denote the boundary layer thickness formed by the respective external flow on the wing or

aerodynamic body.

In the above model the value for ym which depends on the angular (or circumferential)

position on the ramp, the curvature radius and also the height of the blowing slot shall be

calculated.

(3)

1 2

2 , n n

m

m m m

u y yy y

u y y

2

1

2

( )sech ,m m

m m

k y yuy y

u y y

1

2

0,159my

y

u/um

1

2

y

y

my

Valeriu DRGAN 88


(4)

The model constants are

k=0.8814;

We observe that the boundary layer thickness ym is linear dependent with the reference

thickness y1/2 which represents the thickness corresponding to the tangential velocity equal to

half of the maximum local velocity um. Another key point is that the boundary layer

thickness varies exponentially to the circumferential position.

The final equation of the RWR model quantifies the velocity drop across the ramp as a

ratio of the initial blowing velocity:

(5)

We must point out that the local maximum velocity for angular positions immediately

close to the blowing slot are higher than the blowing velocity uj.

This is one of the reasons for which the Coand effect generates a pressure drop, i.e. it increases the dynamic pressure of the jet so that the static pressure must drop below the

atmospheric pressure (especially in fully expanded jets).

Banner gives a rough estimate of the pressure coefficient due to the Coand effect as a function of slot height h and curvature radius R:

(6)

One of the weaknesses of the RWR model is that it only formally includes the h/R ratio

in its equations. This leads to insensitivity to this ratio in determining the reference thickness

for various geometries. In general the RWR model is used for h/R ratios less than or equal to

2%. By substituting the model constants into Eq.4, one can easily obtain:

(7)

In order to improve the existing model, the current study went on to using the existing

experimental data for a variety of h/R geometries encountered in the literature. The highest

h/R ratio encountered is 5.95% which is the limit of the validation for the proposed model.

By means of non-linear regression, each individual h/R case was expressed in the form of an

empirical equation, having its own individual constants. After a satisfactory general equation

form was established i.e. an equation form that provided close matches for each individual

case, we proceeded to express the individual coefficients as a function of h/R. The resulting

equation for the reference thickness in the CEVA model is:

(8)

1

02 1

yx xR k n h

exp Kh h n k h R

27.3 10K [6;7]n

0,55

4,9 9,6m jC

u uh

2Ph

cR

01

2

0.147 0.497 1 x x

y R expR

1 12 22

6,473 /

5,9 10 / 1 ( / ) / 1,3 10

h R Cy h exp k

hh R h R



With the constant k1

(9)

This confirms Rodmans observation that the expression for the reference thickness must be exponential in shape.

Further development of the hereby proposed CEVA model refers to the variation of the

maximum velocity near the ramp:

(10)

Where the circumferential position on the ramp

0C x x (11)

In this case, quadratic expressions were found more suited for both the general equation

and the individual coefficients.

Figure 3 shows extrapolations of the three equations determined for the individual

coefficients. It can easily be seen that the extrapolations follow a natural tendency which

may be an indication that the validity of the equations proposed hereby exceeds the 6% h/R

ratio.

Fig. 3 Extrapolations of the quadratic equations for the three empirical coefficients determined for Eq.10

Another addition to the CEVA model which is unique is the restriction of the maximum

blowing velocity and the maximum momentum coefficient already described in Ref [27].

22

1 [50,65 (0.38 / )]

3,56 101,62 10

1 10 h Rk

1

2 2 2

2 2

2 2

5 2

9,6 10 25,21 396 6,7 10 0,155 68

2,6 10 5,8 10 3,1

m j

h h C h h

R R h R Ru u

C h h

h R R

Valeriu DRGAN 90


In short, the maximum blowing velocity allowed so that the jet will remain attached to

the surface of the cylindrical ramp is based on Lowrys empirical equation for the maximum pressure difference:

(12)

And hence

(13)

where

Rgas is the perfect gas constant (the symbol is changed in order to avoid the confusion

with the geometric radius R of the ramp)

k is the specific heat ratio of the gas

Lastly, another addition to the model can be the empirical equation provided by

Newman Ref [28]for estimating the boundary layer separation which is valid for h/R ratios

less or equal to 10% - which covers the CEVA model.

(14)

3. MODEL VALIDATION

Due to the fact that the CEVA model is entirely based on the interpretation of existing

experimental data, the correlation with the physical measurements is intrinsic.

In Fig. 4 we show the correlation of Eq. 10 with the experimental data from the

literature of specialty Ref [9].

Fig. 4 The correlation of the experimental data with the proposed CEVA model Eq.10

A first comment is that for very low h/R ratios, the CEVA model predicts behaviour

very similar to that of a plane wall jet, i.e. the plotted graph is all but linear in shape.

Secondly, for high h/R ratios, above 7.5%, the tendency of the model is to be hindered,

almost asymptotically.

1

max 3

2 1.41

1 /

k

kjet gas

j

T R ku

M k h R

3

1.4

/atm

P

P h R

/245 391

1 1,125 /sep

h R

h R



This trend makes the model useful even for higher h/R ratios than 6% since the results

given do not fluctuate too much outside the data for which we can be certain it is valid.

Figure 5 shows similar tendencies in the case of the correlation of Eq.8 with the

experimental data.

Therefore the model obtained here is extended from the h/R = 2% in the case of the

Rodman Wood Roberts model to almost 6% and with the prospect of being sufficiently

accurate to up to 10%.

Fig. 5 The correlation of the experimental data with the proposed CEVA model Eq.8

4. CONCLUSIONS

The paper presents a new mathematical model regarding the velocity distribution of a curved

wall jet subject to the Coand effect. The model can be used to accurately predict velocity and hence pressure distribution for a simple Coand flow, however it can be extended to more complex geometries which are the subject of further work. Improvements are made regarding the range for which the CEVA model is validated as opposed to existing models

such as the Rodman-Wood-Roberts model. In the case of CEVA the maximum h/R ratio for

which the experimental data overlap is 6% whereas the RWR model is only validated for h/R

~ 2%. The superior range is given by the explicit introduction of the h/R ratio into the um and

y1/2 equations. Other additions to the model are a restriction for the maximum blowing

velocity uj which is based on Lowrys empirical equation for maximum blowing pressure. This restriction can also be calculated with an empirical equation originally proposed in Ref

[27]. Also an estimation of the separation angle sep based on Newmans work is introduced.

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Valeriu DRGAN 92


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Ceva Model