May 26, 2005 - layer to spanwise oscillating electromagnetic force. Joung-Ho Lee and ..... control of a turbulent channel flow using Lorentz forces,â Phys. Fluids ...
2nd International Symposium on Seawater Drag Reduction Busan, Korea, 23-26 MAY 2005
Response of spatially-developing turbulent boundary layer to spanwise oscillating electromagnetic force Joung-Ho Lee and Hyung Jin Sung (Korea Advanced Institute of Science and Technology) spanwise oscillation. Choi et al. (1998) suggested that the skin friction reduction by spanwise wall oscillation is a result of net spanwise vorticity generated by the periodic Stokes layer over the oscillating wall, which affects the boundary layer profile by reducing the mean velocity gradient within the viscous sublayer. A different explanation of drag reduction was provided by Dhank & Si (1999) that the attenuation in the formation of streamwise streaks by spanwise wall oscillation may disrupt the self sustaining process of turbulence generation. More recently, the DNS studies indicated that the turbulent intensity is reduced by the disruption of the interaction between streamwise vortices and streaks (Choi et al. 2002 and Quadrio & Sibilla 2000). Although the spanwise wall oscillation effectively reduces the skin friction, it is difficult to implement in real applications because of moving wall in the spanwise direction. As an alternative of the wall motions, the use of electromagnetic force is recently employed for low conductivity fluid like sea water (Berger et al. 2000, Pang & Choi 2004 and Breuer et al. 2004). By using the electromagnetic force, it is easy to produce spanwise oscillation by only simply altering the arrangement of electrodes and polarities of the magnets without oscillating the wall. Berger et al. (2000) showed a possibility of reducing the skin friction drag by spanwise oscillating electromagnetic force applied on the entire channel wall. Their numerical studies for a fully-developed turbulent channel flow reported that a maximum drag reduction can be achieved at T+=100. It is not practical in real applications to apply electromagnetic forces on the entire walls, however, a local electromagnetic forcing is necessary for aiming at local drag reduction and local separation delay and so on. In the present study, the effect of local spanwise oscillating electromagnetic force on a turbulent boundary layer was investigated through direct numerical simulations of the spatially-developing turbulent boundary layer. Particular attention was devoted to the spatial evolutions of turbulent boundary layer and the recovery process after the local spanwise oscillation. Two periods of oscillation (T+=100 and 500) were
ABSTRACT Direct numerical simulations were performed to investigate the physics of a spatially-developing turbulent boundary layer flow subjected to spanwise oscillating electromagnetic force. The electromagnetic force beneath the flat plate was locally given with a finite length. A fully implicit fractional step method was employed to simulate the flow. The mean flow properties and the Reynolds stresses were obtained to analyze the nearwall turbulent structure. The skin friction and turbulent kinetic energy can be reduced by the electromagnetic forces. In the absence of the electromagnetic force, the flow eventually relaxes back to a two-dimensional equilibrium boundary layer. INTRODUCTION Realizing flow control system for drag reduction has been a subject of considerable engineering interest because of the high potential benefits, energy saving in particular. Drag is the resistance to the movement that is experienced by something that is moving through a fluid. A large fraction of drag is due to skin friction drag (Gad-el-Hak, 2000) and thus reducing the skin friction drag is the efficient way to reduce the total drag. A recently devised method for skin friction drag reduction is cyclic motions in the spanwise direction imposed by external forcing such as wall oscillation or electromagnetic force. When the oscillation is optimized with appropriate amplitude and frequency, a significant skin friction drag reduction is achieved. Many experimental and numerical studies have been performed to examine the drag reduction by spanwise wall oscillations (Jung et al. 1992, Laadhari et al. 1994, Di Cicca et al. 2002 and Karniadakis & Choi 2003). Jung et al. (1992) firstly showed that the skin friction drag reduction could be achieved by spanwise wall oscillation. In their numerical simulations, a maximum skin friction drag reduction could be obtained at T+=100, where T+ is the nondimensional period of oscillation defined by T+=T(uIJ)2/Ȟ. Many attempts have been made to explain the physical mechanisms responsible for drag reduction by
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chosen for comparison. Time-mean velocities, turbulent intensities and Reynolds shear stresses were analyzed to characterize the dynamic response of turbulent structure to the spanwise oscillation. The budgets of the Reynolds stresses were scrutinized to understand the turbulence statistics. Finally, instantaneous flow visualization techniques were used to observe the response of streamwise vortices and streak structures to the spanwise oscillating forces.
Lx
320T in
computational domain
Ly ReT
300
y
Lz
z 40T in
30T in
turbulent boundary layer
x
& F
& & JuB
U˳
GOVERNING EQUATIONS
N
+
S
a
a
a
wall a
The governing equations for an electrically conducting, incompressible Newtonian fluid are wu u u wt u 0,
1
U
p Q 2 u
1
U
( J u B ),
Figure 1: Schematic diagram of computational domain and configuration of electrodes and magnets.
(1)
(2)
without iteration. Since the implicit decoupling procedure relieves the CFL restriction, the computational time is reduced significantly. The overall accuracy in time is second-order. All the terms are resolved with a second-order central difference scheme in space with a staggered mesh. Details regarding the numerical algorithm are available in Kim et al. (2002).
where u is the velocity vector, ȡ is the fluid density, p is the pressure, ȣ is the kinematic viscosity, J is the electric current density and B is the magnetic flux density. The last term on the right-hand side of Eq. (1) represents electromagnetic force acting on the fluid. The Maxwell equations are directly solved in order to calculate a realistic force distribution.
RESULTS AND DISCUSSION
NUMERICAL METHODS Direct numerical simulations are performed to investigate the physics of a spatially developing turb- ulent boundary layer flow subjected to spanwise wall-normal and spanwise directions, where the correspond-ding mesh size is 513×65×129. The mesh is uniform in the streamwise and spanwise directions, but a hyperbolic tangent distribution is used in the wall-normal direction. The mesh resolutions are ǻx+ § 9.92, ǻymin+ § 0.17, ǻymax+ § 23.86 and ǻz+ § 4.96 based on the friction velocity at the inlet. The computational time step used is ǻt+ § 0.25. Time-dependent turbulent inflow data are provided at the inlet based on the method by Lund et al. (1998). A convective boundary condition at the exit has the form (u/t)+c(u/x) = 0, where c is the local bulk velocity. The no-slip condition is imposed at the solid wall, and the boundary conditions on the top surface of the computational do main are u=U and (v/y) = (w/y) = 0. A periodic boundary condition is applied in the spanwise direction. The governing equations (1) and (2) are integrated in time by using a fractional step method with an implicit velocity decoupling procedure, which has been the coupled velocity components in the convection term are decoupled by the implicit velocity decoupling procedure. Decoupled velocity components are solved
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The configuration of electrodes and magnets that we tested is shown in Fig. 1. Alternating spanwise electrodes and magnets side by side produces an electromagnetic force in the spanwise direction. This configuration is similar to that used by Berger et al. (2000). The array of electrodes and magnets extends from x = 40șin to x = 103șin, where the location of the inlet is defined as x = 0. The array width is 1000 in wall units. The electrode and magnet widths a+ set to 10 in wall units in order to have the greatest effect on the streamwise vortices responsible for high skin friction, where the center is located at approximately y+ = 15 ~ 25 away from the wall. The amplitude of the electromagnetic force is fixed at St = J0B0į/ȡuIJ2 § 85, which corresponds to the optimum frequency St 20S ReW / T suggested by Berger et al. (2000). Here the Stuart number St is defined as the ratio of the electromagnetic force to the inertia force of the fluid. First, we investigate the variations of the mean wall variables along the wall due to local spanwise oscillating electromagnetic force. Streamwise distributions of the time-averaged skin friction coefficient Cf and wall pressure Pw are shown in Fig. 2. The skin friction coefficient Cf, , which is defined as 1 C f W w / UU f2 , is normalized by the free-stream 2
a)
T+=500
+
T =0
Cf
0.006 0.004
+
T =100
2
Pw/UU f
0.002
b)
0.003
Electromagnetic forcing region
+
T =500
0
T+=0
-0.003 0
40
80
+
T =100 120
160
x/Tin
200
240
0 8
vccrms-vccrms,0
4 0 8
wccrms-wccrms,0
4 0 8
-uccvcc+ uccvcc0
4 0 0
40
80
120
160
x/Tin
200
240
280
Figure 2: Contours of the differences of time-averaged rms velocity fluctuations and Reynolds shear stress between forcing and no forcing. The subscript "0" denotes ‚no forcing’. Contour levels are from -0.03Uf to 0.03Uf by increments of 0.002Uf for the velocity fluctuations and from -0.001 Uf2 to 0.001 Uf2 by increments of 0.0001 Uf2 for the Reynolds shear stress.
velocity. As shown, the spanwise oscillating electromagnetic force can reduce the skin friction drag significantly for the optimum period at T+=100. However, the skin friction is decreased and even increased for the high forcing period at T+=500. These results are similar to the turbulent channel simulations by Berger et al. (2000), where the forcing was applied on the entire walls. For T+=100, the skin friction rapidly decreases after the foremost forcing region and reaches a minimum at the end of forcing region. Approximately 20% maximum drag reduction is achieved. As compared with the case of T+=500, no apparent differences are shown in the initial region x/Tin < 70. However, C f grows up to about 10% after x/Tin = 70
where the instantaneous quantity q is decomposed into a time-mean component q , an oscillating component q~ , and a random fluctuating component q cc . The time average is 1 Ttot Lz
q ( x, y )
Ttot
Lz
0
0
³ ³
q ( x, y, z , t )dzdt ,
(14)
where Ttot = NT is the time over which the quantity is averaged and N is the total number of periods. The oscillating component q~ is obtained from the relation
for T+=500. For the response of the wall pressure, adverse pressure gradient occurs first and then favorable pressure gradient is shown after x/Tin = 48. After x/Tin = 70, the pressure gradient is almost constant for T+=100 and slightly adverse for T+=500. In the absence of forcing, the wall pressure is constant in the whole computational region except the end of the computational domain. This may be attributed to the artificial exit boundary condition. After the electromagnetic force, it relaxes back to its two-dimensional equilibrium value. The recovery length is independent of the oscillation periods, at least for the present two forcing periods and its length is about 150Tin. The imposition of spanwise oscillation can lead to periodic variations in the global physical quantities of the flow. Hence, it is necessary to represent each flow quantity as a superposition of three components: q ( s, y ) q~ ( x, y, t ) q cc( x, y, z , t ),
uccrms-uccrms,0
4
280
Figure 2: Streamwise distributions of skin friction coefficient and wall pressure.
q ( x, y , z , t )
8
y/Tin y/Tin y/Tin y/Tin
0.008
q~ ( x, y, t )
where
q ( x, y, t ) q ( x, y ),
(15)
q ( x, y, t ) is the phase average, which is
defined as q ( x, y , t )
1 NL z
N
¦³ n 1
Lz
0
q ( x, y, z , t nT )dz .
(16)
Accordingly, the random fluctuating component q cc is expressed as q cc( x, y, z , t )
(13)
325
q ( x, y , z , t ) q ( x , y , t ) .
(17)
3
uccrms-uccrms,0
x/Tin= 100
4 0 8
2
vccrms-vccrms,0
0 8
uccrms
wccrms
4 1
wccrms-wccrms,0
vccrms
4
uccrms,0
wccrms,0
vccrms,0 0
-uccvcc+ uccvcc0
uccvcc
4 0
uccvcc0
-1 80
120
160
x/Tin
200
240
280
0
20
40
60
y+
80
100
Figure 4: Turbulence intensities and Reynolds shear stress in wall units: T+=100, ; T+=0, .
Figure 3: Contours of the differences of time-averaged rms velocity fluctuations and Reynolds shear stress between forcing and no forcing. The subscript "0" denotes ‚no forcing’. Contour levels are from -0.03Uf to 0.03Uf by increments of 0.002Uf for the velocity fluctuations and from -0.001 Uf2 to 0.001 Uf2 by increments of 0.0001 Uf2 for the Reynolds shear stress.
0.6
0.3
0.5
0.25
2 ½
(ucc ) /U f
+
Figure 3 shows contours of the difference between the rms velocity fluctuations and Reynolds stress under the present forcing T+=100 and the values under the no forcing condition T+=0. It is clear that the spanwise cc , v rms cc oscillating electromagnetic force decreases u rms
«
0.4 x/T = 250 in 0.3
x/Tin= 110
0.2
x/Tin= 100
«
0.1 x/T = 80 in 0
a)
x/Tin= 60
0.2
0.15
2 ½
40
(vcc ) /U f
0
0.1
T =100 T+=0 x/Tin= 250 x/Tin= 110 x/Tin= 100
0.05
x/Tin= 80
0
x/Tin= 60
b)
0.3
c . The streamwise and u ccv cc and increases wcrms velocity fluctuations and Reynolds shear stress respond more quickly to the spanwise oscillating force than the wall-normal and spanwise velocity fluctuations. In the fully-developed turbulent channel simulations (Berger et al. 2000), the rms velocity fluctuations cc , v rms cc and wcrms c ) are reduced. In the present ( u rms study, however, the spanwise velocity fluctuations cc increase initially. This is consistent with the wrms previous finding of Quadrio & Ricco (2003), where the increment of spanwise velocity fluctuations is related with the transient oscillatory motion induced by the electromagnetic force. The initial increase of spanwise velocity fluctuations is attenuated but a small increment is still observed in the end of the forcing region (Fig. 4). Profiles of the rms velocity fluctuations cc , v rms cc and wcrms c ) and Reynolds shear stress ( u rms
0.015
0.25
2 ½
(wcc ) /U f
0.012
«
0.2 x/T = 250 in
0.15
0.009
x/Tin= 110
« «
0.1 x/T = 100 in
0
0
x/Tin= 60
10-2
10-1
x/Tin= 110
0.006 0.003
0.05 x/T = 80 in
c)
x/Tin= 250
2
0 8
-uccvcc/U f
y/Tin y/Tin y/Tin y/Tin
8
100
y/Tin
101
d)
x/Tin= 100 x/Tin= 80 x/Tin= 60
10-2
10-1
100
y/Tin
101
Figure 5: Variations of time-averaged turbulent intensities and Reynolds shear stress. a) u''rms; b) v''rms; c) w'' rms; d)
u ccv cc . um reduction of turbulent kinetic energy is about 20% at the end of forcing region. After the forcing, it returns back to its two-dimensional equilibrium value. The recovery length is about 150șin. A similar reduction of turbulent kinetic energy in the initial region can be seen in the previous results of three-dimensional turbulent flow (Moin et al. 1990, Coleman et al. 1996 and Kannepalli & Piomelli 2000). However, k grows
( u ccv cc ) at several locations are shown in Fig. 5 The profiles of turbulent kinetic energy k u iccu icc / 2 at several locations are shown in Fig. 6 a). 2
Despite the growth of wcc , immediately after the foremost forcing region, the turbulent kinetic energy decreases from its two-dimensional value. The maxim-
326
a)
Reynolds stress ( u cc 2 , v cc 2 and u ccv cc ) are reduced. However, the peak value of spanwise turbulence
0.06
x/Tin= 250
1
0.048
x/Tin= 110
0.8
x/Tin= 110
intensity wcc 2 is slightly enhanced and then finally
0.036
x/Tin= 100
0.6
x/Tin= 100
0.024
x/Tin= 80
0.4
x/Tin= 80
0.012
x/Tin= 60
0.2
x/Tin= 60
decreased. Despite the growth of wcc 2 , the turbulent kinetic energy decreases from its two-dimensional value. After stopping of the electromagnetic force, the flow eventually relaxes back to a two-dimensional equilibrium boundary layer and its recovery length is independent of the oscillation periods.
a1
2
k/U f
streamwise and wall-normal turbulence intensities and x/Tin= 250
0
b) 0
25
50 +
75
100
0 0
25
50 +
y
75
REFERENCES
100
y
Gad-el-Hak, M., Flow control : passive, active flow management, Cambridge university press, 2000.
Figure 6: Variations of turbulent kinetic energy k and structure parameter a1.
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again and becomes lager than the two-dimensional equilibrium value contrary to the present results. To evaluate the efficiency of the eddies in producing the turbulent shear stresses for a given amount of turbulent kinetic energy, we consider the 2
Laadhari, F., Skandaji, L. and Morel, R., “Turbulence reduction in a boundary layer by a local spanwise oscillating surface,” Phys. Fluids A, Vol. 6, 1994, pp. 3218-3220.
2
structure parameter a1 (u ccv cc v ccwcc )1 / 2 / 2k , as shown in Fig.6 b). In the near-wall region, a1 decreases immediately after the electromagnetic force is applied. This trend near the wall appears to be due to a diminishing of Reynolds shear stress u ccv cc despite the decrease of k (Anderson & Eaton 1989 and Coleman et al. 1996). This indicates that spanwise oscillation makes less efficient extraction of shear stress from a given amount of turbulent kinetic energy.
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In the present study, a detailed numerical analysis was performed to delineate the response of a spatiallydeveloping turbulent boundary layer flow subjected to spanwise oscillating electromagnetic force. Statistical descriptions of flow quantities were obtained by performing direct numerical simulations of turbulent boundary layer for two oscillating periods (T + = 100 and 500). The spatial variations of flow quantities along the streamwise direction have been studied. It was found that the skin friction drag is significantly reduced for the optimum period at T+=100. Approximately 20% maximum drag reduction is achieved. However, it is even increased for T+=500 at the end of forcing region. The recovery length is independent of the oscillation periods. For the response of the wall pressure, adverse pressure gradient occurs first and then favorable pressure gradient is shown after the maximum reduction of friction. The
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