**2.1. Mathematical model**

Let us consider the steady two-dimensional flow of an incompressible fluid developed on a cylindrical surface like a jet-sheet. The boundary-layer type equations are written in a curvilinear coordinate system shown in Figure 1. Assuming that the width of the jet slot is small compared to the curvature radius of the cylinder, *R* , the boundary-layer approximations can be applied yielding the simplified equations of motion

$$\frac{1}{r}\frac{\partial V\_{\theta}}{\partial \theta} + \frac{\partial V\_{r}}{\partial r} + \frac{V\_{r}}{r} = 0,\tag{1}$$

$$V\_r \cdot \frac{\partial V\_\theta}{\partial r} + \frac{V\_\theta}{r} \cdot \frac{\partial V\_\theta}{\partial \theta} = \frac{1}{\rho} \cdot \frac{\partial \tau}{\partial r} \,\tag{2}$$

$$
\rho \frac{V\_{\theta}^{2}}{r} = \frac{\partial p}{\partial r} \,\,\,\,\,\tag{3}
$$

where the laminar shear stress is *<sup>V</sup> r* and the turbulent shear stress, where the contribution of the laminar sublayer is neglected (omit the term *<sup>V</sup> r* ), has the form , *<sup>t</sup> V r* with *<sup>t</sup>* the turbulent viscosity, assumed constant in a cross-section like shear layer, i.e., / . *<sup>c</sup>* 

**Figure 1.** Coordinate system and notation.

*t*

102 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**2. Similar solution for a Coanda flow** 

sheet thickness depending on the angle of deviation.

further, more accurate approach of the problem.

thrust, see e.g., [6].

**2.1. Mathematical model** 

Jets are frequently observed to adhere to and flow round nearby solid boundaries. This general class of phenomena, which may be observed in both liquid and gaseous jets, is known as the Coanda effect. In recent years, great interest has been taken in flows deflected by a curved surface. Studying this phenomenon is very important due to the possibility of using the Coanda effect to aircrafts with short takeoff and landing, for fluidic vectoring.

This section deals with the steady two-dimensional, laminar and turbulent flow of an incompressible fluid that develops like a jet-sheet on a cylinder surface, i.e., a Coanda flow [5]. We show that this flow can be approximated well enough by similar solutions for both the laminar and the turbulent regime. Basically we use Falkner-Skan transformations of the momentum equations that can be reduced to one ordinary differential equation (ODE). These solutions are presented in this section for both the laminar and the turbulent flow. The results are given in the form of analytical expressions for the mass flow, thrust and jet-

We also consider the possibility of the thrust augmentation yielded by the fluid entrainment of the jet flow. Thrust vectoring of aircraft which is the key technology for current and future air vehicles, can be achieved by utilizing the Coanda effect to alter the angle of the primary jet from an engine exhaust nozzle. Furthermore, the increased entrainment by the Coanda surface coupled with the primary jet fluid can augment the

The problem considered here is only a crude approximation of the physical phenomenon. However, we believe that the singular solutions that we develop pave the way towards a

Let us consider the steady two-dimensional flow of an incompressible fluid developed on a cylindrical surface like a jet-sheet. The boundary-layer type equations are written in a curvilinear coordinate system shown in Figure 1. Assuming that the width of the jet slot is small compared to the curvature radius of the cylinder, *R* , the boundary-layer

> <sup>1</sup> 0 , *V VV r r r rr*

<sup>1</sup> , *<sup>r</sup> VVV*

> 2 , *<sup>V</sup> <sup>p</sup> r r*

 

*rr r*

 

  (1)

(2)

(3)

approximations can be applied yielding the simplified equations of motion

*V*

The variables in equations (1)-(3) can be made dimensionless, as follows

$$\left| V\_r \right|^\* = \frac{V\_r}{\mathcal{U}\_j}, \left| V\_\theta \right|^\* = \frac{V\_\theta}{\mathcal{U}\_j}, \left| r^\* \right| = \frac{r}{R}, \left| p^\* \right| = \frac{P}{\rho \mathcal{U}\_j^2}, \text{Re} = \frac{\mathcal{U}\_j R}{\nu}, \left| y^\* \right| = \frac{y}{R} = r^\* - 1, \text{ } \nu = \mu \text{ } \left| \rho \right|, \text{ } y^\* \ll 1, \text{ } \mu = \frac{\mathcal{U}\_j}{\mathcal{U}\_j}$$

where *Uj* is the velocity of the jet at the exit of nozzle <sup>0</sup> (assumed constant in crosssection), Re is the Reynolds number based on the cylinder radius and *y* is the radial distance from the cylinder surface, i.e.,*r R* .

The dimensionless continuity equation (1) is satisfied by a stream function, chosen such that \* <sup>1</sup> / *Vr <sup>r</sup>* and \* *V r* . Since \* *y* is much smaller than the unity, it may be neglected compared with the unity in the dimensionless equation (2). Introducing a modelling variable of the form

$$\eta = \text{Re} \cdot y^\* \cdot \frac{c+1}{\sigma} \theta^{(c+1)(a-1)} \tag{4}$$

and with the stream function chosen as

$$
\psi = \theta^{a(c+1)} f(\eta),
\tag{5}
$$

equation (2) can be transformed into the following nonlinear ordinary differential equation:

$$(f''' + af\, f' + \left(1 - 2a\right)f'^2 = 0.\tag{6}$$

Mathematical Modelling and Numerical Investigations on the Coanda Effect 105

4 2 *f ff f* (11)

2 3/2 1/2 6 , *ff ff* (12)

and *f* can be expressed as explicit functions of the parameter *F* and

2

<sup>2</sup> *<sup>f</sup> f F* . (14)

*<sup>f</sup> f FF* (15)

*<sup>f</sup> f FF* (16)

*<sup>f</sup> f FF F* (17)

ranges from 0

(13)

2

4 2 *f f ff f* and

Hence, equation (6) is integrated for the value *a* corresponding to the Coanda flow. Writing

1 1 <sup>2</sup> 0.

Choosing 1/2 *F ff* / equation (12) becomes an equation with

3 1 4 3 2 ln . 2 1

are presented in Figure 2

23 3 1 58 .

3 3 1 14 , <sup>72</sup>

Thus, the problem is completely determined and the solution of equation (11) with given boundary conditions is represented by equations (13)-(17). Calculated values

<sup>4</sup> , <sup>6</sup>

2

*F FF f actg <sup>F</sup> <sup>F</sup>*

We further obtain explicit relations of *F* for the quantities *f* , *f f* , as follows:

3

4

288

The domain of values of *F* is from [0, 1], hence the independent variable

Successively integrating equation (11) with the integrant factors *f* and 3/2 *f* and taking into

account the appropriate boundary conditions firstly gives 1 1 2 2 <sup>0</sup>

equation (6) with *a* 1/4 yields

then

where *f* lim . *f* 

Thus, the quantities

so *f* becomes

to ,[0, ).

of 2 3

*<sup>f</sup>* , *ff f f f f* / ,/ , / and <sup>4</sup> *<sup>f</sup>* / *<sup>f</sup>*

separable variables whose solution is written as

The choice of the constant *a* depends on the boundary conditions. By definition,

$$V\_r^\* = -\frac{1}{r^\*} \cdot \frac{\mathbf{d}\,\nu}{\mathbf{d}\,\theta} \equiv -\theta^{a(c+1)-1} \left[ a(c+1)f + (c+1)(a-1)\eta f' \right] \prime$$

$$V\_\theta^\* = \frac{\partial \nu}{\partial r^\*} = \frac{\partial \nu}{\partial y^\*} = \text{Re}\frac{c+1}{\sigma} \theta^{(c+1)(2a-1)} f' \prime.$$

Integrating equation (6) easily proves that the values *a* 1/3 and *a* 1/2 satisfy the boundary conditions of the free jet flow case and of the boundary layer on a flat plate with zero incident, respectively. For the Coanda type flow considered here, the boundary conditions attached to equation (6) are


Integrating equation (6) from to , with the above conditions, yields

$$-f'' - qf' + \left(1 - 3a\right) \int\_{\eta}^{\wp} f'^2 \mathbf{d} \,\eta = 0.\tag{7}$$

Further integrating equation (7) by means of the integrant factor *f* , yields:

$$\frac{1}{2}f'^2 - af\int\_{\eta}^{\eta} f'^2 \, \mathrm{d}\, \eta + \left(1 - 4a\right) \Big\|\, g' \, \mathrm{d}\, \eta = 0\,\,,\tag{8}$$

where <sup>2</sup> *g f* d . 

Equation (8) written at the wall, i.e., 0 , leads to 0 1 4 d 0. *a gf* Since 0 *gf* d 0, then either 14 0 *a* , or *a* 1/4 . This value satisfies the boundary conditions for the Coanda flow. Thus, the velocity components can be written as

$$V\_r^\* = -\frac{c+1}{4} \theta^{\frac{c+1}{4}-1} \left[1 - 3\eta f'\right],\tag{9}$$

$$V\_{\theta}^{\*} = \text{Re}\frac{c+1}{\sigma} \theta^{-\frac{c+1}{2}} f'.\tag{10}$$

Hence, equation (6) is integrated for the value *a* corresponding to the Coanda flow. Writing equation (6) with *a* 1/4 yields

104 Nonlinearity, Bifurcation and Chaos – Theory and Applications

\*

*r*

conditions attached to equation (6) are

Integrating equation (6) from

where <sup>2</sup> *g f* d .

 

Equation (8) written at the wall, i.e.,

0; 0, 0 0, *f f* (non-slip condition),

: 0, 0, *f f* (condition at the edge).

equation (2) can be transformed into the following nonlinear ordinary differential equation:

\* 1 1

*a c Vr ac f c a f*

1 d 1 11 , <sup>d</sup>

to , with the above conditions, yields

 

0 , leads to

then either 14 0 *a* , or *a* 1/4 . This value satisfies the boundary conditions for the

<sup>1</sup> <sup>1</sup> \* <sup>4</sup> <sup>1</sup> 13 , <sup>4</sup> *c*

\* <sup>2</sup> <sup>1</sup> Re . *<sup>c</sup> <sup>c</sup> V f*

 

1

*<sup>c</sup> V f* 

(7)

 

(8)

0 1 4 d 0. *a gf* 

(9)

(10)

Since

0

*gf* d 0, 

 <sup>2</sup> *f aff a f* 1 3 d 0. 

<sup>1</sup> 2 2d 1 4 d 0, <sup>2</sup> *f af f a gf*

\* <sup>1</sup> 12 1 Re . *<sup>c</sup> c a V f*

Integrating equation (6) easily proves that the values *a* 1/3 and *a* 1/2 satisfy the boundary conditions of the free jet flow case and of the boundary layer on a flat plate with zero incident, respectively. For the Coanda type flow considered here, the boundary

The choice of the constant *a* depends on the boundary conditions. By definition,

*r y*

Further integrating equation (7) by means of the integrant factor *f* , yields:

Coanda flow. Thus, the velocity components can be written as

*r*

 

<sup>2</sup> *f af f a f* 1 2 0. (6)

 

$$f'' + \frac{1}{4} \, f \, f'' + \frac{1}{2} \, f'^2 = 0. \tag{11}$$

Successively integrating equation (11) with the integrant factors *f* and 3/2 *f* and taking into account the appropriate boundary conditions firstly gives 1 1 2 2 <sup>0</sup> 4 2 *f f ff f* and then

$$\left(\hbar f' + f^2 = f\_{\alpha}^{3/2} f^{1/2}\right) \tag{12}$$

where *f* lim . *f* Choosing 1/2 *F ff* / equation (12) becomes an equation with separable variables whose solution is written as

$$
\eta \, f\_{\text{ev}} = 4 \sqrt{3} \, \text{act} \, \text{tg} \left( \frac{\sqrt{3}F}{2+F} \right) + 2 \ln \frac{F^2 + F + 1}{\left(F - 1\right)^2}. \tag{13}
$$

Thus, the quantities and *f* can be expressed as explicit functions of the parameter *F* and so *f* becomes

$$f = f\_{\ll} F^2. \tag{14}$$

We further obtain explicit relations of *F* for the quantities *f* , *f f* , as follows:

$$f' = \frac{f\_{\infty}^2}{6} \left( F - F^4 \right),\tag{15}$$

$$f'' = \frac{f\_{\alpha}^3}{72} \left( 1 - F^3 \right) \left( 1 - 4F^3 \right),\tag{16}$$

$$f''' = -\frac{f\_{\infty}^4}{288} F^2 \left(1 - F^3\right) \left(5 - 8F^3\right). \tag{17}$$

The domain of values of *F* is from [0, 1], hence the independent variable ranges from 0 to ,[0, ).

Thus, the problem is completely determined and the solution of equation (11) with given boundary conditions is represented by equations (13)-(17). Calculated values of 2 3 *<sup>f</sup>* , *ff f f f f* / ,/ , / and <sup>4</sup> *<sup>f</sup>* / *<sup>f</sup>* are presented in Figure 2

**Figure 2.** Variations of terms for the similar solution.

### **2.2. Results**

**Laminar flow.** Since for a laminar flow, / 1, *<sup>t</sup>* then 1 and 0 *c* . So, the components of the velocity become

$$V\_r^\* = -\frac{\theta^{-3/4}}{4} \lbrack f - 3\eta f' \rbrack. \tag{18}$$

$$V\_{\theta}^{\*} = \text{Re}\,\theta^{-1/2} f^{\prime}.\tag{19}$$

Mathematical Modelling and Numerical Investigations on the Coanda Effect 107

 

(21)

(22)

(23)

(24)

(25)

(26)

 

where the first term

, depends only on the curvature wall

. At the origin

0 ,

mechanics, where only the physical size of the body (mass) is taken into account. According to this model, the theoretical jet with given momentum comes out from a zero width slot at

the flow is singular. Equating the momentum of the physical jet with the one theoretically

and further

1/4 0 0 Re *bR f* d .

> 1/4 0 Re . <sup>18</sup> *b f R Q*

> > 2

*b R* Re . 18

1/2 1/4 <sup>18</sup> . Re

0 1 d d .

3/4 , <sup>4</sup>

Next, we calculate the thrust produced by the deflection of the jet with 90 . If *<sup>t</sup> F* is the

d d, *<sup>t</sup> F V y pp y*

 

defined as a measure of the fluid amount involved in flow. The fluid entrainment is an important physical process because it determines the extent of the attachment region of the

> d *<sup>j</sup> A Vy U R*

For the laminar flow, we obtain the dimensionless entrainment parameter as

which shows that the entrainment attains its maximum at the jet origin.

thrust per width unity, then 1 1

*<sup>f</sup> <sup>A</sup>*

2 0 0

 

is a curvature parameter. In [7] an entrainment parameter *A* has been

0

*<sup>b</sup> <sup>f</sup> <sup>R</sup>*

2

3

0 and achieves its mass flow in similar flow conditions at 0

 

 yields 0 2 2 0 d *Ub V y <sup>j</sup>* 

The integral from equation (21) is easily computed yielding

from (20) into (22) leads to

that does not appear in the expression of 0

Coanda effect. The entrainment parameter is defined by

determined at 0

Substituting *f*

This means that *f*

Thus, *f*

 

and the jet characteristics, i.e.,

If the mass flow and the momentum of the jet are given at output ( <sup>0</sup> ) then we are able to compute the value of 0 . Since the boundary layer approximations are not valid near the origin of the jet, this value should be regarded as a virtual origin of the similar motion. Considering the mass flow per length unity of the slot , *Q Ub m j* and *b* the width of the slot, then

$$\mathcal{Q}\_m = \rho \mathcal{U}\_j b = \int\_0^\circ \rho V\_\vartheta dy \Big|\_{\theta\_0} = \rho \mathcal{U}\_j R \theta\_0^{1/4} f\_{\Rightarrow} \theta$$

or

$$
\Theta^{1/4} = \left(\frac{b}{R}\right)\frac{1}{f\_{\phi}}.\tag{20}
$$

Now, let the momentum of the jet in the slot be <sup>2</sup> . *Ub QU j mj* This assumption holds if the velocity does not depend on the length of the slot. This contradicts the assumption that the flows are similar at 0 . The jet model is similar to the material point model of solid mechanics, where only the physical size of the body (mass) is taken into account. According to this model, the theoretical jet with given momentum comes out from a zero width slot at 0 and achieves its mass flow in similar flow conditions at 0 . At the origin 0 , the flow is singular. Equating the momentum of the physical jet with the one theoretically determined at 0 yields 0 2 2 0 d *Ub V y <sup>j</sup>* and further

$$b = R \frac{\text{Re}}{\theta\_0^{1/4}} \prod\_{0}^{\circ} f'^2 \text{d}\,\eta. \tag{21}$$

The integral from equation (21) is easily computed yielding

$$\frac{b}{R} = \frac{\text{Re}}{\mathbb{Q}\_0^{1/4}} \cdot \frac{f\_\alpha^3}{18}. \tag{22}$$

Substituting *f* from (20) into (22) leads to

106 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Figure 2.** Variations of terms for the similar solution.

components of the velocity become

to compute the value of 0

flows are similar at 0

 

**Laminar flow.** Since for a laminar flow, / 1, *<sup>t</sup>*

 

3/4 \* 3 , <sup>4</sup> *V ff <sup>r</sup>* 

\* 1/2 *V f* Re .

origin of the jet, this value should be regarded as a virtual origin of the similar motion.

, *Q U b V dy U R f m j j*

1/4 <sup>1</sup> . *<sup>b</sup> R f*

velocity does not depend on the length of the slot. This contradicts the assumption that the

0

 

0

 

If the mass flow and the momentum of the jet are given at output ( <sup>0</sup>

Considering the mass flow per length unity of the slot , *Q Ub m j*

Now, let the momentum of the jet in the slot be <sup>2</sup> . *Ub QU j mj*

then 1

. Since the boundary layer approximations are not valid near the

1/4 0

. The jet model is similar to the material point model of solid

(18)

(19)

 

(20)

This assumption holds if the

and 0 *c* . So, the

) then we are able

and *b* the width of the

**2.2. Results** 

slot, then

or

$$
\theta\_0 = \left(\frac{b}{R}\right)^2 \frac{\text{Re}}{18}.\tag{23}
$$

Thus, *f* that does not appear in the expression of 0 , depends only on the curvature wall and the jet characteristics, i.e.,

$$f\_{\varphi} = \left(\frac{b}{R}\right)^{1/2} \left(\frac{18}{\text{Re}}\right)^{1/4}.\tag{24}$$

This means that *f* is a curvature parameter. In [7] an entrainment parameter *A* has been defined as a measure of the fluid amount involved in flow. The fluid entrainment is an important physical process because it determines the extent of the attachment region of the Coanda effect. The entrainment parameter is defined by

$$A = \frac{1}{\mathcal{U}\_{\dot{\jmath}}} \cdot \frac{\mathbf{d}}{R \mathbf{d} \,\theta} \Big|\_{0}^{\circ} V\_{\theta} \mathbf{d} \mathbf{y}. \tag{25}$$

For the laminar flow, we obtain the dimensionless entrainment parameter as

$$A = \frac{f\_{\text{cc}}}{4} \cdot \theta^{-3/4} \,, \tag{26}$$

which shows that the entrainment attains its maximum at the jet origin.

Next, we calculate the thrust produced by the deflection of the jet with 90 . If *<sup>t</sup> F* is the thrust per width unity, then 1 1 2 0 0 d d, *<sup>t</sup> F V y pp y* where the first term represents the flow of the momentum and 1 0 90 . Integrating equation (3) yields

$$p\_{\infty} - p = \int\_{r}^{\circ} \rho V\_{\vartheta}^{2} \frac{\mathbf{d}r}{r} = \int\_{\cdot}^{\circ} \frac{\rho \mathbf{U}\_{\cdot}^{2} V\_{\vartheta}^{"\prime 2}}{1 + \mathbf{y}^{\*}} \mathbf{d}\mathbf{y}^{\*} \,\mathrm{d}\mathbf{y}^{\*} \,\mathrm{d}\mathbf{y}$$

Since \* *y* is much smaller compared to the unity, we may neglect it in the denominator.

$$\begin{array}{cccc}\text{Hence,} & \text{the} & \text{integral} & \text{becomes} & \int\_{\eta}^{\eta} \rho \text{J}\_{/}^{2} \text{Re}\,\theta^{-1/4} f^{2} \,\text{d}\eta, & & \text{which} & \text{ leads to} \\\\ p\_{\alpha} - p = \rho \text{J}\_{/}^{2} \text{Re}\frac{f\_{\alpha}^{3}}{18\theta^{1/4}} \Big(1 - F^{3}\Big)^{2}. & & & \end{array}$$

Integrating once, we obtain the second term, which is the contribution of pressure, i.e.,

$$\int\_0^\phi (p - p\_\infty) \mathrm{d}y \Big|\_{\vartheta = \vartheta\_1} = \frac{\rho \mathrm{d}I\_j^2}{2} \mathrm{R}\theta\_1^{1/2} f\_\infty^2 \dots$$

As the total momentum flux is 3 2 1/4 Re , <sup>1</sup> <sup>18</sup> *<sup>j</sup> f U R* the expression of thrust force becomes

$$F\_t = \rho \mathcal{U} I\_f^2 \mathcal{R} f\_\alpha^2 \left[ \frac{\text{Re} \, f\_\alpha}{18 \, \theta^{1/4}} - \frac{\theta\_1^{1/2}}{2} \right]. \tag{27}$$

Mathematical Modelling and Numerical Investigations on the Coanda Effect 109

max 1/3 <sup>1</sup> / 8

<sup>4</sup> *f f* implying that

for

 

~ *x* and

2 *<sup>m</sup>*

*V V* 

max

\* max max

 

\*

the problem is to find the value of

<sup>4</sup> / . <sup>3</sup> *f f FF* For max *f f* / 0.01 it follows that 0.998421 *<sup>F</sup>* and by equation

**Turbulent flow.** For turbulent flows it is first necessary to determine the eddy viscosity . *<sup>t</sup>*

Wall jets are important test cases for turbulence models because they contain a boundary-layer near the wall which interacts with a free shear layer (Figure 3). Thus, they grow much less than free jets. This reduction in the wall jet development is mainly due to the presence of the wall surface where the entrainment by the jet is inhibited on the side nearest to the surface. The velocity fluctuation damping is transmitted to the outer layer and, since the transfer of side momentum component is closely related to the side component of velocity fluctuations, the shear stress and the development of the jet are reduced. Conversely, a relatively high turbulence degree of the outer layer of the wall jet has an effect similar to the turbulence of free jet in the boundary layer case. Usually, the models of turbulence based on turbulent viscosity do not take into account the jet damping at the side wall and in this way, the empirical constants used for the free jet overestimate the development of the jet wall. For example, the

 model with standard constants, given in [8], yields values of the width of the jet augmented by 30 %. Turbulence models based on the shear stress transport equations (6) can correctly predict the wall jet, with standard constants, only if the equations take into account the wall effect on the correlation pressure – strain velocity, see [8], [9]. If this influence is not taken into account the models yield 20% higher values. Use of the turbulence models based on transport equations is however complex and expensive. Hence, in this section (for analytical solution) we are limited to a simple algebraic model, of turbulent viscosity type, which

We assume that the turbulent viscosity for a moderate curvature of the flow is governed by

*t m KV y K V y*

and the mixing coefficient *K* is an empirical constant, same as in the case without curvature.

 

*<sup>m</sup>* is the maximum speed of flow, 1/2 *<sup>y</sup>* is the point of the outer layer where <sup>1</sup> ,

\* \*

(30)

 1/2 Re , *<sup>m</sup>* 1/2

estimates accurately enough the main features of the considered Coanda flow.

the same laws as in the case without curvature. Hence

. Substituting these values in (4) finally yields \* 31,622 3/4 , Re *<sup>f</sup>*

the distance along the cylinder surface, one notices that \* 3/4

is measured from the apparent origin (the point of zero sheet thickness).

Since

\* 1/2

 4/3 4

(13), we have 31.622 *f*

max

where the angle

max \* 1/2 *V x* ~ . 

*k* 

where *V*

Denoting by *x R*

Re , *j l*

*V VV <sup>f</sup> V f UV f <sup>V</sup>*

which max *f f* / 0.01. By equation (15), <sup>2</sup>

By comparison with the non-deflected free jet, we introduce an enhancement factor of thrust defined by the ratio

$$T = \frac{F\_t}{F\_t} \Big|\_{\theta = \theta\_0} = \frac{F\_t}{\rho L I\_f^2 b} = \frac{R}{b} \theta\_1^{-1/4} \left[ \frac{\text{Re} \, f\_\infty^3}{18} - \frac{f\_\infty^2 \theta\_1^{3/4}}{2} \right]. \tag{28}$$

Using equations (22), (23) and (24), *T* becomes

$$T = \left(\frac{\theta\_0}{\theta\_1}\right)^{1/4} - \frac{1}{2} \left[\frac{9\pi}{\text{Re}} + \left(\frac{b}{R}\right)^2\right]^{1/2}.\tag{29}$$

Since 1 0 / 2, 9 / Re 1 and 2 1, *<sup>b</sup> R* one finds that *T* 1, i.e., for the considered

case there is no thrust increase, but only a change in its direction.

It defines the finite thickness of the jet sheet, , the same as the boundary layer, namely the value of *y* where the section 0 has max 0.01. *V V* 

$$\text{Since } \left. V\_{\theta}^{\*} = \frac{V\_{\theta}}{U\_{j}} = \text{Re} \cdot f^{\*1/2}, \quad \frac{V\_{\theta}}{V\_{\theta \text{max}}} = \frac{V\_{\theta}^{\*}}{V\_{\theta\_{\text{max}}}^{\*}} = \frac{f^{\*}}{f\_{\text{max}}^{\*}} \quad \text{the problem is to find the value of } \eta \text{ for } \theta$$

\*

*r y*

 <sup>3</sup> <sup>2</sup> 2 3 1/4 Re 1 .

 *F* 

*r U V*

*r y*

18 *<sup>j</sup>*

As the total momentum flux is

defined by the ratio

Since 1 0

*<sup>f</sup> ppU*

*pp V y*

represents the flow of the momentum and 1 0

2 \*2 2 \*

\* <sup>d</sup> d . 1 *j*

Hence, the integral becomes 2 1/4 2 Re d , *U f <sup>j</sup>*

0

1 0

 

 

Using equations (22), (23) and (24), *T* becomes

 / 2, 9 / Re 1 and

It defines the finite thickness of the jet sheet,

value of *y* where the section 0

case there is no thrust increase, but only a change in its direction.

 has

*t t t j*

*F U b b*

0 1

 

Since \* *y* is much smaller compared to the unity, we may neglect it in the denominator.

 2

d . 2

1/2 2 1

*U R* the expression of thrust force becomes

3 2 3/4

Re . 18 2

1/2

1/4 1

 

Integrating once, we obtain the second term, which is the contribution of pressure, i.e.,

*Uj pp y R f* 

3

2 2 1 1/4 Re . <sup>18</sup> <sup>2</sup> *t j*

By comparison with the non-deflected free jet, we introduce an enhancement factor of thrust

1/2 1/4 <sup>2</sup>

 

0.01.

max

*V V* 

1 9 . 2 Re

*R*

*f*

<sup>1</sup>

2 1/4 Re , <sup>1</sup> <sup>18</sup> *<sup>j</sup>*

*<sup>f</sup> F U Rf*

 

2 1

*<sup>F</sup> <sup>F</sup> <sup>R</sup> f f <sup>T</sup>*

*<sup>b</sup> <sup>T</sup>*

2 1, *<sup>b</sup> R* 

   90 . Integrating equation (3) yields

which leads to

(27)

(28)

(29)

one finds that *T* 1, i.e., for the considered

, the same as the boundary layer, namely the

which max *f f* / 0.01. By equation (15), <sup>2</sup> max 1/3 <sup>1</sup> / 8 <sup>4</sup> *f f* implying that 4/3 4 max <sup>4</sup> / . <sup>3</sup> *f f FF* For max *f f* / 0.01 it follows that 0.998421 *<sup>F</sup>* and by equation (13), we have 31.622 *f* . Substituting these values in (4) finally yields \* 31,622 3/4 , Re *<sup>f</sup>* where the angle is measured from the apparent origin (the point of zero sheet thickness). Denoting by *x R* the distance along the cylinder surface, one notices that \* 3/4 ~ *x* and

max \* 1/2 *V x* ~ . 

**Turbulent flow.** For turbulent flows it is first necessary to determine the eddy viscosity . *<sup>t</sup>* Wall jets are important test cases for turbulence models because they contain a boundary-layer near the wall which interacts with a free shear layer (Figure 3). Thus, they grow much less than free jets. This reduction in the wall jet development is mainly due to the presence of the wall surface where the entrainment by the jet is inhibited on the side nearest to the surface. The velocity fluctuation damping is transmitted to the outer layer and, since the transfer of side momentum component is closely related to the side component of velocity fluctuations, the shear stress and the development of the jet are reduced. Conversely, a relatively high turbulence degree of the outer layer of the wall jet has an effect similar to the turbulence of free jet in the boundary layer case. Usually, the models of turbulence based on turbulent viscosity do not take into account the jet damping at the side wall and in this way, the empirical constants used for the free jet overestimate the development of the jet wall. For example, the *k* model with standard constants, given in [8], yields values of the width of the jet augmented by 30 %. Turbulence models based on the shear stress transport equations (6) can correctly predict the wall jet, with standard constants, only if the equations take into account the wall effect on the correlation pressure – strain velocity, see [8], [9]. If this influence is not taken into account the models yield 20% higher values. Use of the turbulence models based on transport equations is however complex and expensive. Hence, in this section (for analytical solution) we are limited to a simple algebraic model, of turbulent viscosity type, which estimates accurately enough the main features of the considered Coanda flow.

We assume that the turbulent viscosity for a moderate curvature of the flow is governed by the same laws as in the case without curvature. Hence

$$\boldsymbol{\nu}\_{t} = \boldsymbol{K} \cdot \boldsymbol{V}\_{\partial m} \cdot \boldsymbol{y}\_{1/2} = \nu \,\mathrm{K} \,\mathrm{Re}\,\boldsymbol{V}\_{\partial m}^{\*} \boldsymbol{y}\_{1/2}^{\*}\,\tag{30}$$

where *V<sup>m</sup>* is the maximum speed of flow, 1/2 *<sup>y</sup>* is the point of the outer layer where <sup>1</sup> , 2 *<sup>m</sup> V V* 

and the mixing coefficient *K* is an empirical constant, same as in the case without curvature.

**Figure 3.** Similar velocity profiles.

**Figure 4.** Governing laws for the wall jet flow.

It is easily seen that there are similar solutions of the motion equations (1), (2) and (3), if / *t* has the form

$$\left|\nu\_{\sharp}\right\rangle\left|\nu=\sigma\theta^{\mathfrak{c}}\right.\tag{31}$$

Mathematical Modelling and Numerical Investigations on the Coanda Effect 111

*m V f V f* 

\* 2/3 / 3 3, *V ff <sup>r</sup>*

\* 4 Re 2/3 ; <sup>3</sup> *V f*

296 ; <sup>27</sup> *b R*

1/3 3

2/3 ; <sup>3</sup>

<sup>1</sup> 1 1 2

 

> *f*

*R*

2.37 *<sup>b</sup> <sup>f</sup> <sup>R</sup>*

*<sup>f</sup> <sup>A</sup>*

1/3 0 1 1 0

*<sup>b</sup> <sup>T</sup>*

 \* 1/2 max *y VV* 0.075 /2 ; 

 


, it follows that 0.90396 *<sup>F</sup>* and 14.8

0

'

max

'

 

2/3

;

which yields *c* 1/ 3. Since the rate of increase

it follows that 0.90396 *F*

which leads to the

1/2 <sup>4</sup> d /d Re , <sup>3</sup> *y R*

(32)

(33)

(34)

(35)

(36)

(38)

(37)

 For

. Substituting in (4) we

0.5

). With these data we can now specify the development

of equation (4) must be written as \*

148 *f*

which is used to compute 14.8

estimation Re




max

obtain the final result

*m V f V f* 

0.5



 ~ /, *y* 

in width is 1/2 d / d 0.075 *y x* , see [6], we obtain 1/2

(from the condition

*f*

of the considered self-modelled flow (Figure 4):


In the present model the viscosity does not take into account the outer intermittent flow, so that the assumption of constant value in the cross section leads to velocity profiles identical to those of laminar flows, though the general configuration of the flow development is different. Suppose in this case that a linear flow develops, then the dimensionless coordinate of equation (4) must be written as \* ~ /, *y* which yields *c* 1/ 3. Since the rate of increase in width is 1/2 d / d 0.075 *y x* , see [6], we obtain 1/2 1/2 <sup>4</sup> d /d Re , <sup>3</sup> *y R* which leads to the

estimation Re 148 *f* (from the condition ' ' max 0.5 *m V f V f* it follows that 0.90396 *F*

which is used to compute 14.8 *f* ). With these data we can now specify the development of the considered self-modelled flow (Figure 4):


110 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Figure 3.** Similar velocity profiles.

**Figure 4.** Governing laws for the wall jet flow.

has the form

/ *t* 

It is easily seen that there are similar solutions of the motion equations (1), (2) and (3), if

/ . *<sup>c</sup>*

In the present model the viscosity does not take into account the outer intermittent flow, so that the assumption of constant value in the cross section leads to velocity profiles identical to those of laminar flows, though the general configuration of the flow development is different. Suppose in this case that a linear flow develops, then the dimensionless coordinate

(31)

*t* 

$$V\_r^\* = -\theta^{-2/3} / \Im \left[ f - \Im \eta f' \right]\_{\prime} \tag{32}$$

$$V\_{\theta}^{\*} = \frac{4}{3} \frac{\text{Re}}{\sigma} f' \theta^{-2/3};\tag{33}$$


$$
\theta\_0 = \frac{296}{27} \frac{b}{R};
\tag{34}
$$


$$f\_{\circ} = \frac{3}{2.37^{1/3}} \left(\frac{b}{R}\right)^{2/3};\tag{35}$$


$$A = \frac{f\_{\text{cc}}}{3} \theta^{-2/3};$$
 
$$\text{(36)}$$


$$T = \left(\frac{\theta\_0}{\theta\_1}\right)^{1/3} \left[1 - \frac{1}{2}\frac{b}{R}\frac{\theta\_1}{\theta\_0}\right] < 1\tag{37}$$


$$\boldsymbol{y}\_{1/2}^{\*} = 0.075 \,\theta \quad \left(\boldsymbol{V}\_{\theta} = \boldsymbol{V}\_{\theta \max} \,/\, \mathcal{D} \right);\tag{38}$$


$$\frac{V\_{\theta m}}{\mathcal{U}\_j} = \left(18.5 \frac{b}{\theta \mathcal{R}}\right)^{2/3};\tag{39}$$

Mathematical Modelling and Numerical Investigations on the Coanda Effect 113

model [15] and (4) Menter

SST model, a close match of

SST model to sensitize to the

SST turbulence

model (see

Reynolds averaged Navier-Stokes simulations (RANS) with different turbulence models have been employed in order to compute the two-dimensional turbulent wall jet flowing around a circular cylinder: (1) Spalart and Allmaras (SA - one turbulence model equation)

The particular configuration shown in Figure 5 is considered cylindrical. The wall jet properties have been reported by Neuendorf and Wygnanski [17] and provide the means to evaluate the simulation results (diameter *d* = *2R =* 0.2032 m, nozzle height *b* = 2.34 mm and

The computational grid used for these investigations consists of 900 x 220 nodes. For the turbulence models used in these calculations the laminar sublayer needed to be resolved. The *y* values of the wall-next grid points were between 0.4 and 1, and the *x* values were between 50 and 300. The grid resolution in the jet was between 40 and 180 times the local Kolmogorov length scale. A fully developed channel velocity profile was prescribed at the

For some of these turbulence models the jet-velocity decay and jet-half-thickness versus the streamwise angle are plotted in Figure 6. The jet-half-thickness (*y*1/2) represents the thickness where the jet velocity (*Uj*) is half of the maximum jet velocity (*Um*) through the same section.

the jet-velocity decay with the measured data was achieved. However, even with this

Figure 7). Since the predicted half-thickness (*y*1/2) is small for all models, the normalized velocity profiles do not match the experimental velocity profiles neither in the mild pressure

One weakness of the eddy-viscosity models is that these models are insensitive to streamline curvature and system rotation. Based on the work of Spalart and Shur [18] a modification of

model were presented in Figures 6, 7 and 8, respectively. The results are close to the experimental data up to about 120 degrees. For larger values, the development of the jet was

SST model, the separation location was slightly closer to the experimental

and Spalart-Allmaras models were used, the jet remained attached to

nozzle inflow (no near field), with a medium turbulence. The ambient was quiescent.

model was used in combination with the *k*

The shape of the normalized velocity profiles is best predicted by the *k*

model, the downstream development of the jet-half-thickness was poorly predicted.

model [14], (3) Wilcox *k*

 SST model [16]. The predictions yielded by the simulations were compared to available experimental measurements from the literature. The surface curvature enhances the near-wall shear production of turbulent stresses and is responsible for the entrainment

of the ambient fluid which causes the jet to adhere to the curved surface.

[13], (2) Launder and Spalding *k*

jet-exit velocity *U*j = 48 m/s).

data. When the *k*

curvature effect.

poorly predicted.

region, nor in the adverse pressure region.

the cylinder for more than 260 degrees (see Figure 8).

the production term has been derived, which allows the *k*

The results obtained with the corrected (curvature correction – *c.c.*) *k*

When the *k*

For the *k*

*k* 


$$\text{arg}\_{1/2}^\* \sim \infty \quad V\_{\theta m}^\* \sim \infty^{-2/3}, \quad \text{x} = \text{R}\theta. \tag{40}$$

In many applications that use boundary layer control by tangential blowing, the solid surface downstream of the blowing slot is strongly curved and, in this case, the prediction of the jet involves separation and a more accurate knowledge of the flow (radial and tangential pressure - velocity profiles) which can be done with CFD methods.
