Chapter V. Flow Control Applications
1.2. C μ definition
To take into account both steady and unsteady blowing cases, the original momentum coef-ficient Cμ definition from Poisson-Quinton [47] (coming from experimental empirical findings) has been modified in refs [29][51] as:
𝐶𝐶𝜇𝜇 =ρ𝑗𝑗𝑆𝑆𝑗𝑗 < 𝑈𝑈𝑗𝑗2>𝑡𝑡 12ρ∞𝑆𝑆𝑟𝑟𝑟𝑟𝑟𝑟𝑈𝑈∞2 =
ρ𝑗𝑗𝑆𝑆𝑗𝑗𝑈𝑈𝑟𝑟𝑟𝑟𝑟𝑟2
12ρ∞𝑆𝑆𝑟𝑟𝑟𝑟𝑟𝑟𝑈𝑈∞2 (2)
where ρj is the actuator flow density, Sj the sum of each actuator orifice surface, Uj(t) is the blow-ing velocity, < >t is the time-averaging operator, Urms is the root-mean square value of the veloci-ty, ρ∞ is the freestream density, Sref is the controlled area in 2D (controlled span multiplied by the chord length) and the total wing area in 3D and U∞ is the freestream velocity.
As the numerator and the denominator are homogeneous to a momentum flux, this coefficient should be more properly called momentum flux coefficient rather than momentum coefficient. One can also remark that the numerator is homogeneous to a force and consequently Cμ has the same non-dimensionalisation as the lift coefficient. This is the reason why the lift gain is sometimes plotted as function of CD (drag) + Cμ since the jet momentum could have been used to propel the aircraft.
The RMS value of the velocity Urms for a square signal and for any DC value is:
𝑈𝑈𝑟𝑟𝑟𝑟𝑟𝑟2 =1𝑇𝑇∫ 𝑈𝑈0𝑇𝑇 2(𝑑𝑑)𝑑𝑑𝑑𝑑 =1𝑇𝑇∫0𝐷𝐷𝐷𝐷∗𝑇𝑇𝑈𝑈2(𝑑𝑑)𝑑𝑑𝑑𝑑 since U(t)=0 for t > DC*T density is not constant. This expression of the momentum coefficient in the pulsed blowing case which I have proposed during the AVERT European project is the one now used in all European projects with active flow control: JTI Clean Sky, AFLoNext, JTI Clean Sky 2, etc.
A lot of things can be learnt just by looking at the momentum coefficient definition in eq.
(3). First, for a fixed qm value, the lower the DC value, the higher the momentum coefficient. So, in pulsed blowing mode, it seems interesting to decrease DC below the classical value of 0.5, i.e.
to blow with a high velocity on a short duration. This is because the blowing velocity is squared in eq. (2). In practice, to keep the time-averaged mass flow rate qm value constant when DC decreas-es, it means that the velocity during blowing is increased and so the compressed air feeding stag-nation pressure. At first sight, it seems appealing for a given Cμ value to decrease the mass flow consumption on an aircraft by decreasing the duty cycle. But, to obtain higher air feeding pres-sure, since the compressor power is equal to the product of the volumetric flow rate (decreased)
by the stagnation power (increased), the power consumption will be the same. Now, if there is no compressor because the bleed air comes from the engines, one can decrease the duty cycle to de-crease the time-averaged mass flow rate. The peak velocity during the blowing phase will inde-crease until reaching M = 1 and the flow in the slot will be shocked.
If the stagnation pressure Pi and temperature Ti of the jet are known, for example by a measurement in the actuator cavity, the time-averaged Mach number M can be computed by in-verting the following equation:
This equation has two solutions for a given mass flow rate: one subsonic and one superson-ic. Since a nozzle is not used generally, the right solution is the subsonic one.
Then, the time-averaged velocity 𝑈𝑈� can be computed by the equation:
2
Or, if the stagnation pressure and temperature are unknown, by making the rough approxi-mation ρ𝑗𝑗 ≈ρ∞:
This rough approximation is all the more inaccurate that the jet velocity approaches the son-ic one.
For the pulsed blowing case with DC = 50%, equation (3) becomes:
2
For the constant blowing case, equation (3) becomes:
2
It means that for the same time-averaged mass flow rate, the momentum coefficient of the pulsed blowing case with DC = 50% is twice the one of the constant blowing case. In other terms, the pulsed blowing slot will require a time-averaged mass flow rate √0.5 = 30% smaller than a con-tinuous blowing slot to have the same effect on the flow. It has been checked experimentally in the PhD thesis of T. Chabert that the lift gain with pulsed blowing at DC = 50% was twice the one with continuous blowing even if the time-averaged mass flow rates were equal. This can be un-derstood by the fact that reducing DC leads to a higher peak velocity during the blowing phase and since the jet velocity is squared in the numerator, the momentum coefficient is larger.
For a synthetic jet with a sinusoidal jet velocity, the RMS value of the velocity Urms is:
𝑈𝑈𝑟𝑟𝑟𝑟𝑟𝑟2 =1
𝑇𝑇 � 𝑈𝑈2(𝑑𝑑)𝑑𝑑𝑑𝑑 = 1 𝑇𝑇
𝑇𝑇 0
�(𝑈𝑈𝑝𝑝𝑟𝑟𝑝𝑝𝑝𝑝sin 𝑆𝑆𝑑𝑑)2
𝑇𝑇 0
𝑑𝑑𝑑𝑑
𝑈𝑈𝑟𝑟𝑟𝑟𝑟𝑟2 =𝑈𝑈𝑝𝑝𝑟𝑟𝑝𝑝𝑝𝑝2 𝑇𝑇 �
1 − cos 2𝑆𝑆𝑑𝑑
2 𝑑𝑑𝑑𝑑 =
𝑇𝑇 0
𝑈𝑈𝑝𝑝𝑟𝑟𝑝𝑝𝑝𝑝2 𝑇𝑇 �
𝑑𝑑 2 −
1
4𝑆𝑆 sin 2𝑆𝑆𝑑𝑑�0 𝑇𝑇
=𝑈𝑈𝑝𝑝𝑟𝑟𝑝𝑝𝑝𝑝2 2
So, the momentum coefficient for a synthetic jet is equal to:
𝐶𝐶𝜇𝜇 = ρ𝑗𝑗𝑆𝑆𝑗𝑗𝑈𝑈𝑟𝑟𝑟𝑟𝑟𝑟2 12ρ∞𝑆𝑆𝑟𝑟𝑟𝑟𝑟𝑟𝑈𝑈∞2 =
ρ𝑗𝑗𝑆𝑆𝑗𝑗𝑈𝑈𝑝𝑝𝑟𝑟𝑝𝑝𝑝𝑝2
ρ∞𝑆𝑆𝑟𝑟𝑟𝑟𝑟𝑟𝑈𝑈∞2 (6)
When comparing eq. (4) and (6), one can remark that a pulsed blowing jet with DC = 0.5 has the same momentum coefficient as a synthetic jet so they should have the same effect on the flow if the momentum coefficient is the right non-dimensionalised parameter:
In the Cμ definition from eq. (2), the blowing angle does not appear. Yet, one can expect that a tangential or a normal blowing slot will not have the same effect on the flow. To study the effect of the blowing angle on a continuous blowing case, a database coming from an optimisation in the PhD thesis of M. Meunier [41] has been reused. Fig. V.4 shows the slotless flap configura-tion with a blowing slot at the flap knee. It consists in 129 RANS computaconfigura-tions with three param-eters: the blowing velocity, the blowing angle (with respect to the local normal) and the actuator location. Fig. V.5 shows the lift coefficient as function of the three parameters.
Fig. V.4: Slotless flap configuration.
Fig. V.5: Lift coefficient for the 129 samples as function of slot location and blowing angle and velocity.
Now, if the lift coefficient is plotted as function of the momentum coefficient and slot location in Fig. V.6, one can remark two things. First, if the slot is located downstream of the flap knee, the lift is lower because the slot is located in the separation. Secondly, the higher the mo-mentum coefficient, the higher the lift except for some odd points which are surrounded by a black circle and for which lift is much lower because in eq. (3), the blowing angle does not ap-pear.
Fig. V.6: Lift coefficient as function of momentum coefficient and slot location.
A modified Cμ is then proposed to take into account the blowing angle:
2 2
2 1
) cos (
∞
∞
=
U S U C S
ref j j j
ρ
α ρ
m (7)
where α the blowing angle with respect to the local tangent. This definition is only valid for the case of momentum injection (tangential blowing with continuously blowing slots or pulsed blow-ing/zero-net-mass-flux slots at high frequency). In this mode, only the tangential component of the velocity is useful for control in this case. For the case of normal blowing slots, the physical
mechanism is different and involves momentum mixing with the external flow by spanwise roll-ers. Fig. V.7 shows the lift coefficient as function of the slot location and the modified Cμ. The odd points have disappeared since they were corresponding to normal blowing slots which do not contribute to increase the boundary layer momentum and now correspond to low Cμ values.
Fig. V.7: Lift coefficient as function of modified momentum coefficient and slot location.
The next question is: how varies lift with the momentum coefficient? On the same test case as in Fig. V.4, lift is plotted as function of the unmodified momentum coefficient. One can remark that lift increases linearly with Cμ up to the full reattachment of the flow on the flap. This is the separation control part of the curve. Then, if one continues to increase Cμ, there is the circu-lation control mode for which lift increases with the square root of Cμ.
Fig. V.8: Lift coefficient as function of momentum coefficient.
When looking at the numerator of the Cμ definition in eq. (3), one can remark that if the bleed air is coming from a reservoir, increasing the jet Mach number will result in a decrease of the jet density and an increase of the jet velocity. Then, the question is: for given stagnation pres-sure Pi and temperature Ti from a reservoir, for which jet Mach number the momentum coeffi-cient is maximised? The numerator of eq. (3) can be rewritten as:
M S rate, the momentum flux and the jet power are plotted as function of the jet Mach number in Fig.
V.9. As expected, in a pipe for example, the mass flow rate is maximum for M = 1. The momen-tum flux is, as said previously, maximum for M = 2. So, it seems more interesting to have a supersonic jet which would require the presence of a nozzle in the blowing slots. Nevertheless, the power increases continuously so for practical reasons, in order to not have to manufacture a noz-zle shape in a thin slot, the solution M = 1 is chosen.
In flow control, another parameter which has to be chosen is the slot width. Of course, from eq. (3), the larger the orifice area Sj, the larger the momentum coefficient. But, for a given mass flow rate when bleed air is taken from the engine, how to choose the slot width? One can wander if the slot width has to be chosen with respect to the local boundary layer thickness for example. The effect of the momentum coefficient being very high, it is better to maximise it by choosing the slot width in order to have M = 1 at the slot exit even for low speed applications. Of course, in practice, there are some manufacturing constraints and the slot cannot be very thin (<
0.3 mm is difficult).
Fig. V.9: Mass flow rate qm (in red), momentum flux ρV2S (in blue) and blowing power ρV3S (in black) as function of the Mach number for given stagnation pressure and temperature.
Fig. V.10 shows numerical results of the effect of the momentum coefficient on the max-imum lift gain for active flow control on a two-element laminar airfoil with continuous blowing through a slot at the leading edge. Lift increases linearly with Cμ up to a lift gain of 25% where it reaches saturation. One can remark that all curves are not superimposed but that the 0.5 mm wide slot requires twice the Cμ of the 0.25 mm slots. Moreover, the curves do not cross the null lift gain horizontal axis for Cμ = 0 which means that a minimum value of Cμ is required to have a positive gain. If Cμ is too low, lift decreases. It is well-known that one has to blow at least at the local ve-locity to have a positive effect. If one injects a too low momentum in the boundary layer, it will separate earlier.
Power
Fig. V.10: Lift coefficient as function of the momentum coefficient.
As explained in the definition of Cμ, this coefficient is coming from experimental empiri-cal findings. Another way, more theoretiempiri-cal, is to perform a momentum balance on a control vol-ume around the slot (see ref. [37] for more details). In this way, a new non-dimensionalised pa-rameter CBLC is found. In the subsonic jet case, the expression is:
−
=C VR
CBLC 1
m 1 (8)
where VR is the ratio of the jet velocity to the local external flow one. So, the CBLC coefficient will be equal to zero for VR = 1. Moreover, this coefficient enables to take into account the negative effect of blowing at a too low velocity ratio on the flow (CBLC < 0 for VR < 1). Fig. V.10 is replot-ted as function of the CBLC coefficient in Fig. V.11. With this coefficient, the lift gain is equal to zero for CBLC = 0.
In the supersonic jet case, the expression is:
−
− +
=
∞
∞ C VR
U S
S p C p
ref j j
BLC
1 1 2
1
) (
2 m
ρ (9)
where pj is the jet static pressure and p is the local static pressure.
slot width 0.5mm upper side
slot width 0.25mm with nozzle M=1.4
slot width 0.25mm with nozzle M=2
Fig. V.11: Lift coefficient as function of the CBLC coefficient.
Since with a tangential slot, the goal is to inject momentum in the boundary layer, it is interesting to look at the streamwise evolution of the peak velocity in the boundary layer along the chord in the case of a continuously blowing slot at the leading edge. Fig. V.12 shows that, with continuous blowing (black curve), the peak velocity decreases inversely as the square root of the streamwise coordinate. This is in agreement with the wall jet theory even if there is a strong ad-verse pressure gradient on the airfoil in the present case. One can remark that the velocity at the airfoil trailing edge (x = 0.5) is close to the freestream velocity (U∞ = 68 m.s-1). For a synthetic jet (red curve), the velocity decreases faster. To construct a design tool to estimate roughly the mass flow rate requirement on a aircraft for active flow control at the leading edge, a perspective of this study could be to use a boundary layer code with in entry the pressure distribution in order to roughly estimate the required blowing velocity like it has been done by Carrière et al. in ref. [10].
Fig. V.12: Peak boundary layer velocity as function of the streamwise coordinate for a continuous jet (in black) and a synthetic jet (in red).
So, to summarize this section, it has been shown that pulsed blowing allow reducing the mass flow rate by 30% to reach the same Cμ value. It has also been shown that a pulsed jet and a synthetic jet with the same peak velocities have the same Cμ value. In summary of all Cμ
defini-tions given in this section, a proper definition of Cμ (to take into account both steady and unsteady blowing, the effect of the blowing angle and the velocity ratio) would be: