The first study into spanwise wall oscillation was by Jung et al. [1992], who inves- tigated a turbulent channel flow withReτ = 200. The oscillation was applied using
the formula: ww =Wmsin 2π T t (2.1)
where the spanwise velocity at the wall ww is determined by the maximum wall
velocity, Wm, the period of oscillation, T, and the time, t. The maximum wall
velocity was fixed at Wm = 0.8 and a range of parameters were studied; T+ =
25,50,100,200 and 500. The optimal value ofT+ was found to be≈100 where the wall shear stress was shown to reduce by 40%. The rms fluctuations were shown to be reduced by the oscillation and the peaks moved away from the wall. The time evolution of the streamwise wall shear stress showed a large oscillation with greater time periods.
The understanding was furthered by Baron and Quadrio [1996], in which the optimum parameter ofT+= 100 was taken from the previous study and simulations
were performed with amplitudesWm= 0,0.25,0.5,0.75,1. The drag reduction was
shown to increase with increasing velocity amplitude achieving a maximum at just over 40%. The power spent was also calculated and smallest amplitudeWm= 0.25
was shown to give net power saving in the order of 10%. A thorough study of the turbulent statistics was performed for theWm= 0.75 case in order to understand the
flow physics. It was seen that the velocity and vorticity rms values were reduced by the control method, but foru0+ (scaled by local units) this reduction was limited to the near-wall region. The turbulent kinetic energy budget showed a decrease in near wall production, hence the location of maximum was increased. A large decrease in the dissipation and viscous diffusion were also seen in the near wall region. Looking at the third and fourth moments it was found that the streamwise skewness was increased near the wall and the maximum location was moved from the wall (in the no-control case) to y+ = 5−10. The wall normal component of skewness showed a region of large negative skewness in the region 5< y+<40 with a local minima introduced at y+ = 15. It was seen in the flatness that the u and w components exhibited their maximum values at slightly higher y+, where as v had larger wall values.
The effect of wall oscillation was also studied in pipe flow by Nikitin [2000] at
Re= 4000 (corresponding to Reτ = 180). The study agreed that using oscillation
frequency ω+ = 0.06 gave the maximum drag reduction. Using the relationship
ω = 2Tπ, this is analogous to a time period of T+ ≈ 100. W+ = 3,6 and 9 were studied and relaminarisation was seen for the larger amplitude. The strong similarity between the results from pipe and channel flow showed that the large wall curvature had little effect on the behaviour of near-wall structures.
Choi et al. [2002] performed DNS of channel flow atReτ = 100,200 and 400
and turbulent pipe flow was simulated with Reτ = 150. Although a view on the
effect of Reynolds number is studied the domain size is quite small and therefore the large scale structures may not be sufficiently independent in space. Also the grid
spacing used is quite large and may not successfully capture the small scales. The paper showed that for T+ = 150, in pipe flow, the mean spanwise velocity profile corresponded well the the laminar stokes solution. The drag reduction was again shown to increase withWm+ and plateau atWm+≈10, while the optimal T+ is also seen at around 100. Interestingly, these features were seen to be very similar, when the parameters are scaled by wall units, in the higher Reynolds number case, with the overallDR achieved reducing as the value of Reτ is increased.
The aforementioned paper, Choi et al. [2002], also looks into the physics of wall oscillation flow, plotting conditionally averaged flow-fields based on Q = 1
2
Ω2ij−Sij2>0. These fields, shown at different points in an oscillation period, illustrates the idea that when the vorticity is acting in a direction opposed to the wall motion the high speed fluid is moved below the low-speed fluid. Conversely, when the vortex and wall move in the same direction the high-speed fluid is moved away from the vortex and the influence of the low speed fluid is weakened. One of the major contributions of this paper is the introduction of the scaling parameter defined by: S+= a + 5y + d A+Re0.2 τ = √2 T+ln Wm+ Wth+ exp −y¯+ r π T+ , (2.2)
whereWth+is chosen as 0.5 and ¯y+= 5. Note thata5is the acceleration aty+= 5. It is then suggested that drag reduction scales with this parameter, using the relation
DR= 1000S+2+50S+. This parameter is studied in later work and will be discussed further in section 5.2.7.
The initial response to the wall oscillation at Reτ = 200 was explored by
Quadrio and Ricco [2003] via DNS. A range of forcing parameters were studied with 50< T+<200 for fixedWm+ = 18, and 3< Wm+<27 with fixedT+= 125. A small initial delay in the reduction of drag is noted and attributed to the wall-normal distance of the streak/structure locations. This height causes an offset between in
the spanwise velocity felt at the wall an thisylocation due to the propagation of the wall motion into the flow. The delay in the reaction of drag to the application of the forcing is lengthened with increasingT+ or a reduction in Wm+. ForWm+= 18 and
T+ = 125 a good agreement of the spanwise velocity profile is shown between the laminar solution and DNS results. Studying the initial transient for the same forcing parameters shows an initial decrease in u0 followed by an increase, before settling to a steady state. v0,u0v0 and the production termP11 follow quantitatively similar behaviour with a slightly delayed response in the wall normal velocity fluctuation. A large near wall peak is initially generated in w0, before a reduction occurs and a fully developed state is established. The production termP33 is initially zero, but becomes large in the temporal transient before returning to zero.
Quadrio and Ricco [2004] studied the effects of wall oscillation on a turbulent channel flow atReτ = 200. A thorough parametric study was undertaken consisting
of 37 simulations spanning the rangeT+= 0−750 and Wm+ = 0−27. A maximum drag reduction was seen at the location (100,27) in (T+, W+
m)-space. For any given
Wm the maximum drag reduction was found in the rangeT+ = 100−125, and by
increasing Wm the DR is shown to increase at a decreasing rate. One interesting
calculation shown here is that of, not only the power saving Psav but also that of
the power required for the oscillationPreq. These are defined by:
Psav= ∂U0 ∂y − ∂U ∂y (2.3) Preq= ∂W ∂y W (2.4)
Here, U0 represents the space averaged mean of the stream wise velocity from the no-control case, and h·i represents the time averaging procedure. The percentage net power saving is then calculated as %Pnet = %Psav+ %Preq, where %Preq is a
percentage of the wall shear stress from the uncontrolled case and %Psav =DR. It
period ofT+= 125. A net power saving was achieved forT+>70 (whenWm+ = 45) andWm+<7 (whenT+= 125).
Using the simulation results theSparameter scaling toDRis revisited, using a linear scaling. The scaling is confined to oscillations withT+≤150, and the best correlation is found when the ¯y+ = 6.3 and Wth+ = 1.2. This Wth+ value is justified as it is of the order of the turbulent fluctuations. The scaling is then shown to follow the equationDR = 131S+−2.7 and will be discussed further in section 5.2.7. TheDR prediction is plotted against T+ showing an over estimation for large T+. The optimal values, however, are assumed to be predicted correctly and an analytic expression is found to calculateTopt+.
After performing DNS atReτ = 173, Huang and Xu [2004] studied the trans-
port of Reynolds stresses. A single simulation was performed with wall oscillation parameters Wm+ = 15 and T+ = 90, giving a drag reduction of 36%. It is shown analytically that the shear caused by the wall oscillation, dWdy , acts directly on the hu0w0i,hv0w0i and
w02
terms. The results showed that the w0rms and production term P33 initially increased slightly and were subsequently decreased, whereas a monotonic in a manor which is dependent on the decrease was seen in the pres- sure strain term S33. The pressure strain for the wall-normal component is also decreased, reducing thevrms0 . This is amplified by ∂U∂y to reduce the production of the Reynolds shear stressP12=−
v02∂U∂y, diminishing the skin friction.
Ricco and Quadrio [2008] builds upon the previous work in Quadrio and Ricco [2004], using the same domain at Reτ = 200, though the number of grid points in
y was extended to 160. Three further controlled simulations were performed at
Reτ = 400, with Wm+ = 12 and T+ = 30,125,200. The plane-averaged, spanwise
momentum equation is presented:
∂w¯+ ∂t+ = ∂2w¯+ ∂y+2 − ∂v0w0+ ∂y+ ,
where ¯w+is the average of the spanwise velocity over a wall normal plane. The last term is shown to greatly decrease atWm+= 18 andT+= 125, due to the oscillation. This implies an independence ofw to the streamwise flow, justifying the use of the laminar Stokes’ solution in the calculation of S+. The %Psp is also approximated
and an analytic expression for %Psav is presented. One result of note is that of
the Reτ = 400 simulations. It is shown that, at the higher Reynolds number, a
lower level ofDR is achieved for the three cases studied. It is also seen that there is a larger decrease inDR of the oscillations with larger time periods, T+. This is thought to explain the lower level of drag reduction found in experimental data.
Touber and Leschziner [2012] performed DNS at both Reτ = 200 and 500.
At the lower Reynolds number 5 forcing parameters were studied with Wm+ = 12 fixed and, at high values of T+ an increase in drag was observed. The value of
T+ = 100 and 200 were investigated atReτ = 500. The correlations of streamwise
velocityRuu, in the spanwise direction were shown to indicate an effect of the forcing
on the near-wall streaks. The first minimum in the streamwise velocity correlation effectively disappeared when the optimal forcing was applied, corresponding to an increased influence of larger scale structures. The near-wall streaks were visualised and seen to be angled by the oscillation. This streak angle was calculated and its pdf, computed over the oscillation period was considered to be ‘close to bimodal’. The phase of the oscillation of the streak angle was close to that of the shear angle at y+ = 10 and the variation of this phase was said to be small in the near-wall region. A conditionally averaged streak was found and the effect of the Stokes’ layer was illustrated. This spanwise phase-averaged velocity was compared to that of the laminar solution and as in previous study was shown to have good agreement at
T+ = 100. However, for larger periods a substantial difference in this property is observed. LES was also performed at Reτ = 1000 and the drag reduction of the
various Reynolds numbers were compared to the S+ parameter. This highlighted the Reynolds number effect and the fact that this was not currently incorporated
into the scaling parameter. Other statistical properties of the flow were shown including fluctuations PDFs, TKE budgets, enstrophy variation and the spectral density functions.
Recently, Agostini et al. [2014] have published a study of wall oscillation at
Reτ = 1000. Although results ofT+= 100 were shown, the main focus was around
the T+ = 200 case with Wm+ = 12. The results support the previous study of Touber and Leschziner [2012] that points of high skin-friction and streak strength throughout the cycle correspond to high, slowly varying stokes strain. A hysteresis in the drag over the period is attributed to the near-wall velocity skewness. The fluctuations of enstrophy and dissipation show strong similarities over the forcing cycle, and are both shown to have qualitative correlation with the drag.
Cimarelli et al. [2013] looked into the effects of spanwise wall oscillations applying an assortment of different waveforms. It was seen that the sinusoidal forcing remains optimal for T+<100 and obtains the optimal power saving. The waveforms were broken into different modes whose effects were analysed individually. A waveform was then rebuilt to find a waveform that was approximated it give a greater power saving than the sinusoidal wave. This result was confirmed via DNS.