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Magnetic effects on the velocity and thermal fields in the 2D incompressible flow around a cylindrical body
M. Douak* and Z. Aouachria
Laboratory of Applied Energetic Physics (L.A.E.P.), University of Batna, Algeria
Abstract - The fluid separated control of the boundary layer around a body finds numerous applications in varied aeronautic configurations. Because, the ability of this control is to avoid this phenomenon, it represents an interest subject for investigations, although it is subject to many studies from various angles. This operation leads to improve the lift level, and to reduce the noise and the drag force, which are generated by the aeronautic machines. Then, these three parameters present the evident economical and technological stakes. This work presents an analysis of a boundary layer magneto hydrodynamics developing on the wall of a cylindrical body. The simulation of the flow was carrying out, by a suitable change of variables making it possible to remove the existing singularity at the origin of the boundary layer and to standardize the co- ordinates perpendicular to the wall, in a rectangular field of integration. The study carries out clearly the influence of the Reynolds numbers and the interaction magnetic parameter on the boundary layer and its characteristic parameters and makes, in evidence, their effects on the separate flow, thermal and hydrodynamic profiles.
Résumé - Le contrôle du décollement du fluide de la couche limite d’un écoulement autour d’un corps trouve de nombreuses applications dans différentes configurations de l’aéronautique. Parce que l’habilité de ce contrôle est d’éviter ce phénomène, il représente un intéressant sujet d’investigation, bien qu’il soit un objet, de beaucoup d’études sous différents angles. Cette opération conduit à améliorer le niveau de la portance et de réduire la force de traînée et le bruit, qui est généré par les machines aéronautiques. Alors ces trois paramètres présentent les évidents enjeux économiques et technologiques. Ce travail présente une analyse d’une couche limite magnéto hydrodynamique se développant sur la paroi d’un corps cylindrique. La simulation de l’écoulement est présentée à l’aide d’un changement de variables souhaitables permettant d’éliminer la singularité existante à l’origine de la couche limite et de standardiser la coordonnée perpendiculaire à la paroi du corps, dans le domaine rectangulaire d’intégration. L’étude met bien en évidence l’influence du nombre de Reynolds et le paramètre d’interaction magnétique sur la couche limite et de ses paramètres caractéristiques. Elle présente ainsi les effets sur le décollement de l’écoulement et les profils thermique et dynamique.
Keywords: Boundary layer - Magneto hydrodynamic - Flow separation - Thickness of displacement - Constraint viscous - Thermal field.
1. INTRODUCTION
The fluid separation control of the boundary layer around a profile finds numerous applications in varied aeronautic configurations. Because, the ability of this control is to avoid this phenomenon, it represents an interest subject for investigations, although it is subject to many studies from various angles, Yakovlev [1], Favier et al. [2], Weie et al.
[3]. This operation leads to improve the lift level, and to reduce the noise and drag
* douak_2007@yahoo.fr _ aouachria2001@yahoo.fr
force, which are generated by the aeronautic machines. Then, these three parameters present the evident economical and technological stakes, Shatrov [4].
Several studies have, up to now, been managed on the experimental plan to test activator different efficiencies able to control the fluid separation phenomena as Weie et al. [5] and Henoch et al. [6]. Most of these studies concern the active methods using the mobile walls as for example that which reported by Modi [7], the aspiration, thermal and acoustic methods D.C. Me Cormick [8], Chang [9], Collins [10] or electromagnetic methods M.Gad-el-Hak [11].
Numerous studies have been focalized on how to generate electric current around a body. There are two approaches to create an electric current near a body. At first, in a system of conductive, the current is supplied to the fluid through electrodes in direct contact with the fluid. Secondly, in a system of induction, the current locally is caused by an alternating magnetic field inside the body.
For the first time control of the boundary layer to a wall map was proposed by Gailitis and Lielausis [12]. Our work aims to study magnetic field effects on the stationary 2-D hydrodynamic and thermal flow of an incompressible fluid around a cylindrical body.
By using electromagnetic forces, the profile of the boundary layer is changed in accordance with an exponentially with its highest linear stability and well-known, which may even lead to a reduction positive energy, force dragging at the wall. Note that this idea has not yet been considered so far, neither one nor the other in terms of non-linear stability of the boundary layer profile nor in terms of experimentation on the cylindrical body.
This work presents a numerical study of a boundary layer magneto hydrodynamics growing on the wall of a cylindrical obstacle. The numerical simulation of the flow was carried out by an appropriate change of variables, Gőrtler [13] and Shlichting [14].
2. MATHEMATICAL ANALYSES
The work aims to study electric E and magnetic field B effects on the stationary 2D hydrodynamic flow of an incompressible fluid around a cylindrical body. The electromagnetic terms would appear in the governing equations through the Laplace force:
B ) B U E ( . B
JΛ = σ + Λ Λ (1)
We know that, in perfect fluid, this force is conservative and then has not action on the streamline of the flow. If we consider a real fluid, only the part:
B ) B U E (
. + Λ Λ
σ becomes no conservative and it modifies the flow [15].
Accordingly, the numerical study that we go to develop here, will concern a flow of boundary layer type submissive only to a magnetic field applied perpendicularly to a plan of flow.
Then we consider an incompressible 2D dimensional flow fluid around a cylindrical body in the magnetic field, B, oriented perpendicularly to the flow as shown in the figure 1.
By employing the boundary layer approximations and similarly variables below:
υ
∫
=
ξ x
0
e(x).dx u
1. ) x
( ; y
2 ) u
y , x
( e
ξ
= υ
η (2)
And these transformations
ue
F = u ;
∂ η ξ ∂ + υ ν
ξ
∂ ξ
= ∂ −
F x 2 2
V x
1
; Te
= T
θ (3)
Fig. 1: Schematic problem We obtain the dimensionless governing equations:
= ρ −
+ σ ξ
− υ
η
∂
∂ ξ υ
η −
∂ θ
∂ υ α ξ η −
∂ θ
∂ υ + ξ ξ
∂ θ
∂ ν
= ρ −
− σ ξ −
∂
∂ υ
η +
∂
∂ ν
− ξ η
∂
∂ υ + ξ ξ
∂
∂ ν
=
− νξ
η +
∂
∂ ν + ξ ξ
∂
∂
ν ∞
0 ) 1 F T ( c
u B d
u d T c
u F F
T c 2
u
2 V u
2 F u u
0 ) 1 F B ( ) 1 F u ( u
F 2
u V V
2 u F F
u
0 F r . r 2 u 2 V 2
u F u
2 e p
2e e 2
e p
3e 2
e p
4e
2 2 2 e2 2e
e2
2 2 e e
2 2 2 2 e
2 e e
2 2e
e2
(4)
This system of equations is composed into two parts:
The first one is for (ξ ≠0)
( )
=
− γ
− γ
+
η
∂
− ∂ η
− θ η + θ ξ
∂ θ ξ ∂
=
−
−
− β η + η −
∂ + ∂ ξ
∂ ξ ∂
= η +
∂ + ∂ ξ
∂ ξ∂
ξ
0 1 F I F 2
m d Ec
d Pr
1 d Vd F
2
0 ) 1 F ( kI ) I 1 F d (
F d V F
F F 2
0 V F
2 F
2 2 2
2
2 2
2 (5)
And the second part is in the stagnation point (ξ= 0,ue =0)
η =
− θ η θ
=
−
−
− η +
η −
= η +
d 0 d Pr
1 d Vd
0 ) 1 F ( 2I ) 1 1 F d (
F d d
F Vd
0 d F
V d
2 2
2 2 2
(6)
With the boundary conditions:
0 ) 0 , (
F ξ = ; V(ξ,0)=0 ; θ(ξ,0)=0 1
) , (
F ξ ηe = ; V(ξ,ηe)= 0 ; θ(ξ,ηe)=1 (7) Where:
ξ
∂
∂
= ξ
βξ e
e u u
2 ; γ = 4(1−cos(ϕ)) ; k =1+ cos(ϕ) ;
α
= υ Pr ;
cp ρ
= λ
α ;
e 2 T cp Ec = u∞ ;
ρ ∞
= σ u
R
I B2
The viscose constraint is given by this expression bellow:
0 2e F 2 u
=
η
η
∂
∂ ξ
= ρ
τ (8)
3. NUMERICAL STUDY
The numerical solution of the equations is based on the finite differences. The method is consists to discrete transforming domain of the calculus in some elements (Fig. 2).
Fig. 2: Grid of the study domain
p j
) 1 i ( w j ) 1 i ( e ) 1 j ( i n ) 1 j ( i s j
i (a F a F a F a F Sa)/ a
F = − + + + + + − + (9)
j i j
) 1 i ( j
i ) 1 j (
i V ( 2i )F ( 2i )F
V + = + − ∆η + + ∆η− ∆η (10)
where:
η
∆
= −
p 22
a ;
η + ∆ η
− ∆
= 2
1 V
as 2 ij ;
η + ∆ η
−∆
= 2
1 V
an 2 ij ;
) F i (
ae= ij ; aw=(−iFij);
ρ υ
= σB2
H ;
− υ ∆ξ υ
ξ
= ∆
βξ ∞
2
ei r
i r u 2 ue i 2
β − − ∆ξ −
= ξ H(F 1)
e u
i ) 2 1 F (
Sa 2 ij
i 2j
i ; uei=2u∞sin(θ)i;
and for ξ=0.
p )
1 j ( n ) 1 j ( s
i (b F b F Sb)/ b
F = − + + + (11)
j j
i ) 1 j
( V F
V + = − ∆η (12)
where
η + ∆ η
− ∆
= 2
1 V
bs 2 j ;
η + ∆ η
−∆
= 2
1 V
bn 2 j ; bp =bn + bs;
− − −
= I(F 1)
2 ) 1 1 F (
Sb j2 ij ;
The calculus code is valued by the experiments data reported by Weie et al. (2003) and the results obtained are, qualitatively, in good agreement with the work of Weie (1998).
4. RESULTS AND DISCUSSIONS
The figure 3-a and 3-b show that the Reynolds number changes the position of the separate flow fluid on the body surface. We can remark that the separate point goes to back of the cylindrical body (104°
→
107°). The effect of the magnetic field on the separate flow is shown on the figure 4.a- Re=1.8×105 a- I=1.0
b-Re=1.2×105 b- I=1.85
c- Re=1.2×105 c- I= 2.5
Fig. 3: Reynolds effects on the separate flow
Fig.4: Effect of the magnetic interaction parameter, I , on the separated flow,
for a fixed value of Re
We have plotted the variation of dimensionless velocity component, F , for 105
3
Re= × and different values of magnetic interaction parameter, I. We can observe, on the figure 4-a, that the separate flow is localized to the polar coordinate ϕ=101 ° when I=1, whereas for I=1.85, this phenomenon persists but it was displaced to the position which is localized by the polar coordinate ϕ=120° for I=2.5, (Fig. 4-b).
This phenomenon vanishes in this position and comes in to the position ϕ=122° for I =2.5. Then the increasing of magnetic field magnitude pushes the separate point to back of cylinder and we can, from any value of the parameter I , to avoid it.
We remark in figure 4, that the separate point, for a given field magnetic intensity value B , has tendency to be push back towards the downstream when the velocity began to increase.
But at any value of this velocity, the separate point tends to take again its former position. This Phenomenon is observed only in one direction when the field magnetic intensity increases.
The pressure gradient in the boundary layer is plotted on the figure 5. It changes differently in two adjacent regions independontly to the magnetic field. The effect of this magnetic field is shown on the pressure gradient magnitude and on the region expanses. We note clearly this phenomenon, on the figure 6 for Re=1.2×105 and different values of the magnetic interaction parameter I . The pressure gradient decreases on the first region which is situated over the interval (0 - 60°) and increases on the second region situated on the different ranges which are functions of the magnetic field. In fact, we observe on the figure 5, that the region where the pressure gradient increases pass from (45° - 90°) for I= 0 to (60° - 165°) for I=2 and it becomes zero for different positions ϕ.
Fig. 5: Magnetic effect on the pressure gradient
This study shows also that the thermal field is influenced by both Reynolds number and magnetic interaction parameter. We have plotted on the figure 6, the dimensionless temperature evolution in different suctions of the boundary layer and we can show different effects of the Reynolds, Prandtl numbers and parameter I.
We can observe the region near the stagnation point the temperature is high than in the back region of cylinder. Generally we remark that the increasing Re , parameter I and Pr induce a heating of the boundary layer, (Fig. 6).
Finally, we present, in the figure 7, the viscose constraint evolution in function of the Reynolds number and the magnetic parameter values.
On this figure, we can show that there is a fixed value of the viscose constraint for all Reynolds numbers and its maximum moves to downstream of cylinder when the magnetic parameter increases. The fixed value of τ is localized about at ϕ=100°. In presence of the magnetic field, we observe this same phenomenon but its position moves to upstream in the location situate at ϕ=94°, (Fig. 8-a, 8-b, 8-c and 8-d).
a- b- c- Fig. 6: Thermal profiles in the boundary layer
a- Reynolds effect on the temperature profile; b- Magnetic effect on the temperature profile; c- Prandtl on the temperature profile
Fig. 7: Reynolds effect on the viscose constraint
a- I=0.5 b- I =1
c- I=1.5 d- I= 2 Fig. 8: Reynolds and magnetic field effects on the viscose constraint
5. CONCLUSION
The numerical simulation of magneto- hydrodynamic boundary layer developed in this study permits first to eliminate the singularly point located at the origin riper and to transform curvilinear domain to a rectangular domain.
The study carries out clearly, in one side, the influence of the Reynolds numbers and the magnetic parameter on the separate flow, temperature and velocity profiles and in the other side, the evolution of the viscose constraint evolution in function of the Reynolds number and the magnetic parameter values.
The remarkable information about this constraint is its fixed value for all Reynolds numbers and its maximum moves to downstream of cylinder when the magnetic parameter increases.
The calculus code is valued by the experiments data reported by Weie et al., (2003).
The results obtained are, qualitatively, in good agreement with the work of Weie, (1998).
NOMENCLATURE
V Dimensionless velocity σ Electrical conductivity, Ω-1
u Velocity, m/s υ Kinematics viscosity, m2/s
v Velocity, m/s ρ Mass density, kg/m3
ue Velocity, m/s η Dimensionless space
Coordinate
x, y Space coordinate, m ξ Longitudinal coordinate,
kg2/m2.s2
R Radius of cylinder, m α thermal diffusivity, m2/s
B Magnetic field, T λ Thermal conductivity, W/m.K
T Temperature, K δ Boundary layer thickness, m
P Hydrodynamic pressure, Pa δ* Displacement of B, I thickness Cp Pressure coefficient, W/m.K δ** Thickness of the momentum
τ Viscose constraint, N/m2 θ Dimensionless temperature Cp: Specific heat of the fluid at constant pressure, J/kg.K
REFERENCE
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[7] V.J. Modi and F. Mokhtarian, ‘Moving Surface Boundary Layer Control as Applied to 2-D Airfoils’, AIAA paper, pp. 89 -96, 1989.
[8] M. Cormick, ‘Boundary Layer Separation Control with Directed Synthetic Jets’, AIAA paper, 2000.
[9] P.K. Chang, ‘Control of Flow Separation’, Hemisphere, Washington, DC.
[10] F.G. Collins, ‘Boundary Layer Control on Wings Using Sound and Leading Edge Serrations’, pp. 1975 – 1979, 1976.
[11] M. Gad El-Hak, ‘Flow Control, Passive, Active and Reactive Management’, Cambridge University Press, Cambridge, UK, 2000.
[12] A. Gailitis and O. Lielausis, ‘On the Possibility to Reduce the Hydrodynamics Drag of a Plate in an Electrolyte’, Applied Magneto Hydrodynamics, Vol. 13, pp. 143 – 146, 1961.
[13] H. Görtler, ‘A New Series for the Calculation of Steady Laminar Boundary Layer Flows’, Journal of Mathematical Mechanics, Vol. 6, pp. 1 – 66, 1957.
[14] H. Shlichting, ‘Boundary Layer Theory’, University of Braunschweig, Germany Former Director of the Aerodynamishe Versuchsanstalt Gőrttingen, 1979.
