The vortex shedding produces an unsteady flow in the wake that can be characterized by the root-mean-square (rms) magnitude of the velocity fluctuations. Figure 3(c) shows that the streamwise rms velocity exhibits a double-peaked distribution reaching a maximum value of 0.42*U*∞ at {*x*/*D*, *y*/*D*} = {1.67, ±0.47}. The transverse rms velocity exhibits a single-peaked distribution with a maximum value of 0.60*U*∞ at *x*/*D* = 2.22 on the wake centreline as shown in Fig. 3(d). The maxima in the rms velocities are approximately 5% higher than those measured in the cylinder wake at *Re* = 2150 (Konstantinidis & Balabani, 2008). This observation suggests that the mechanism of vortex formation remains basically unaltered over this *Re* range so long as the boundary layer on the cylinder surface remains laminar

Simulations were carried out at different Reynolds numbers in the range *Re* = 60 − 180 and the global flow characteristics were determined. One of the key parameters in cylinder wakes is the mean drag coefficient, *CD*, which is indicative of the mean energy dissipated in the fluid. Table 1 shows the variation of the mean drag coefficient. The table includes the contribution of the skin friction *CDv* and pressure *CDp* components to the total force. The skin friction drag reduces monotonically but non-linearly with increasing Reynolds number as might be expected. On the other hand, the pressure drag increases with increasing Reynolds number. As a result, the total drag exhibits a minimum at *Re* = 150. Results are also shown from another numerical study using a high-order spectral element code (Henderson, 1995). The maximum difference of the total drag between the present study and that of Henderson is 1%. Figure 4 shows the variation of the fluctuating lift coefficient and the Strouhal number as a function of the Reynolds number. Both variables increase as the Reynolds number increases. The solid line in Fig. 4(a) is an empirical fit to experimental data for the sectional lift coefficient

> 2 901/2

*St* = 0.2663 − 1.019√*Re*. (13)

where = (*Re* − 47)/47. The above equation is valid in the range *Re* = 47 − 190. The present simulations predict a lift coefficient which is slightly above the line representing the empirical relationship. Although small, this difference can be attributed to three-dimensional effects inevitably present in experiments and, in particular, the initiation of slantwise vortex shedding (Gerrard, 1966). Norberg (2003) also provided an empirical relationship for the

It should be pointed out that the Strouhal number refers to the frequency at which vortices are shed from one side of the cylinder. The present results agree fairly well with the empirical

(12)

Table 1. Viscous, pressure and total drag for different Reynolds numbers.

*C*� *L* = 30 +

Strouhal number in the same range *Re* = 47 − 190,

relationship even though it is based on experimental data.

until separation.

provided in Norberg (2003),

Fig. 3. Time-averaged flow characteristics for *Re* = 150.

outside separation, the back pressure coefficient is given by (Griffin & Hall, 1991)

$$C\_{pb} = \frac{p\_b - p\_{\infty}}{\frac{1}{2}\rho U\_{\infty}^2} = 1 - K^2 \tag{10}$$

where *K* = *Us*/*U*∞ is the velocity ratio at separation. The maximum velocity magnitude outside separation is 1.36*U*∞ at a distance 0.23*D* from the surface. Figure 3(b) shows the distribution of the mean pressure where a back pressure coefficient of *Cpb* = −0.95 is obtained, compared to a value of −0.85 computed from the above equation. The mean pressure coefficient attains a minimum value of −1.36 on the surface of the cylinder at *θ* = 85◦ which, coincidently, is exactly equal to the velocity ratio slightly further from the surface. The point of time-averaged flow separation is located at *θs* = 112◦. This compares well to the value of 114◦ computed from the following empirical relationship

$$
\theta\_8 = 101.5 + 155.2 \text{Re}^{-1/2}.\tag{11}
$$

The above equation was suggested by Wu et al. (2004) who conducted soap-film experiments and numerical simulations under 2D flow conditions.

6 Will-be-set-by-IN-TECH

0

*y/D*

outside separation, the back pressure coefficient is given by (Griffin & Hall, 1991)

*Cpb* = *pb* − *p*∞ 1 2 *ρU*2 ∞

where *K* = *Us*/*U*∞ is the velocity ratio at separation. The maximum velocity magnitude outside separation is 1.36*U*∞ at a distance 0.23*D* from the surface. Figure 3(b) shows the distribution of the mean pressure where a back pressure coefficient of *Cpb* = −0.95 is obtained, compared to a value of −0.85 computed from the above equation. The mean pressure coefficient attains a minimum value of −1.36 on the surface of the cylinder at *θ* = 85◦ which, coincidently, is exactly equal to the velocity ratio slightly further from the surface. The point of time-averaged flow separation is located at *θs* = 112◦. This compares well to the value of

The above equation was suggested by Wu et al. (2004) who conducted soap-film experiments

2

4

0

*y/D*

2

4

*x/D* -2 0 2 4 6 8 10 -4

0 0.1 0.2 0.3 0.4 0.5 0.6

*x/D* -2 0 2 4 6 8 10 -4

= 1 − *K*2 (10)

*θs* = 101.5 + 155.2*Re*−1/2. (11)

(d) Transverse rms velocity

(b) Mean pressure

*x/D* -2 0 2 4 6 8 10 -4

*x/D* -2 0 2 4 6 8 10 -4

Fig. 3. Time-averaged flow characteristics for *Re* = 150.

114◦ computed from the following empirical relationship

and numerical simulations under 2D flow conditions.

(c) Streamwise rms velocity

(a) Streamwise mean velocity

0 0.1 0.2 0.3 0.4

0

*y/D*

2

4

0

*y/D*

2

4

Table 1. Viscous, pressure and total drag for different Reynolds numbers.

The vortex shedding produces an unsteady flow in the wake that can be characterized by the root-mean-square (rms) magnitude of the velocity fluctuations. Figure 3(c) shows that the streamwise rms velocity exhibits a double-peaked distribution reaching a maximum value of 0.42*U*∞ at {*x*/*D*, *y*/*D*} = {1.67, ±0.47}. The transverse rms velocity exhibits a single-peaked distribution with a maximum value of 0.60*U*∞ at *x*/*D* = 2.22 on the wake centreline as shown in Fig. 3(d). The maxima in the rms velocities are approximately 5% higher than those measured in the cylinder wake at *Re* = 2150 (Konstantinidis & Balabani, 2008). This observation suggests that the mechanism of vortex formation remains basically unaltered over this *Re* range so long as the boundary layer on the cylinder surface remains laminar until separation.

Simulations were carried out at different Reynolds numbers in the range *Re* = 60 − 180 and the global flow characteristics were determined. One of the key parameters in cylinder wakes is the mean drag coefficient, *CD*, which is indicative of the mean energy dissipated in the fluid. Table 1 shows the variation of the mean drag coefficient. The table includes the contribution of the skin friction *CDv* and pressure *CDp* components to the total force. The skin friction drag reduces monotonically but non-linearly with increasing Reynolds number as might be expected. On the other hand, the pressure drag increases with increasing Reynolds number. As a result, the total drag exhibits a minimum at *Re* = 150. Results are also shown from another numerical study using a high-order spectral element code (Henderson, 1995). The maximum difference of the total drag between the present study and that of Henderson is 1%.

Figure 4 shows the variation of the fluctuating lift coefficient and the Strouhal number as a function of the Reynolds number. Both variables increase as the Reynolds number increases. The solid line in Fig. 4(a) is an empirical fit to experimental data for the sectional lift coefficient provided in Norberg (2003),

$$C\_L' = \left(\frac{\epsilon}{30} + \frac{\epsilon^2}{90}\right)^{1/2} \tag{12}$$

where = (*Re* − 47)/47. The above equation is valid in the range *Re* = 47 − 190. The present simulations predict a lift coefficient which is slightly above the line representing the empirical relationship. Although small, this difference can be attributed to three-dimensional effects inevitably present in experiments and, in particular, the initiation of slantwise vortex shedding (Gerrard, 1966). Norberg (2003) also provided an empirical relationship for the Strouhal number in the same range *Re* = 47 − 190,

$$St = 0.2663 - 1.019\sqrt{Re}.\tag{13}$$

It should be pointed out that the Strouhal number refers to the frequency at which vortices are shed from one side of the cylinder. The present results agree fairly well with the empirical relationship even though it is based on experimental data.

∞

Figure 5 shows the frequency of vortex shedding in terms of the Strouhal number, *St* = *fvsD*/*U*∞, as a function of the excitation frequency in terms of the reduced velocity. If there was no interaction between the imposed perturbations and vortex shedding, the Strouhal number would remain constant at the value indicated by a horizontal dashed line. However, the Strouhal number is affected for all the perturbation frequencies considered. For a range of reduced velocities, vortex shedding locks on to the perturbation frequency as indicated by the solid line. For low reduced velocities, the Strouhal number gets higher than its unperturbed value in the natural wake, and vice versa for high reduced velocities. The two cases marked

Bluff Body Aerodynamics and Wake Control 71

Figure 6 shows the instantaneous vorticity distributions for two different reduced velocities together with the variation of the drag and lift forces at three instants over approximately half vortex shedding cycle (cf. Fig. 2). For both cases, single vortices are shed alternatively from each side of the cylinder in a fashion similar to that observed for steady flow. However, the details of vortex formation are different. For example, the peak vorticity values and the size of the vortices is increased in harmonically perturbed flow due to wake resonance. Furthermore, the longitudinal distance between shed vortices varies linearly with the reduced velocity. These changes in the wake patterns are accompanied by considerable changes in the magnitude of the mean and fluctuating forces acting on the cylinder. The unsteady in-line force shows a fifty-fold increase in magnitude whereas the transverse increases by 2.5 times compared to their counterparts in steady flow. The in-line force in unsteady flow contains

inertial aerodynamic components due to added mass and pressure gradients, i.e.,

*FM* = 1 4

*ρπD*2*CM*

where *CM* = 1 + *CA* is the inertia coefficient and *CA* the added mass coefficient. Under the conditions examined here, the inertial components of the in-line force dominate over the viscous component. As a consequence, the in-line force is almost in phase with the flow acceleration d*U*/d*t*. On the other hand, the transverse force is not affected by inertial components since there is no perturbation in that direction. In Fig. 6, the waveform of the transverse force deviates markedly from a pure harmonic for both reduced velocities and there is quite a difference between them reflecting the effect of perturbation frequency on *CY*.

d*U*

d*t* (15)

Fig. 5. Variation of the Strouhal number *vs.* reduced velocity.

by open symbols will be examined in more detail below.

Fig. 4. Flow parameters as a function of Reynolds numbers in steady flow. Symbols show the present results while the lines indicate empirical relationships.

Overall, the results presented here for a range of Reynolds numbers in the laminar unsteady wake regime for steady incident flow compare favourably with experimental data and other numerical simulations which provides confidence in the numerical method employed. The simultaneous presentation of the vortex patterns in the wake and fluid forces on the body has revealed effectively the connection between them.

#### **4. Circular cylinder in harmonically perturbed flow**

The wake response to harmonic perturbations in the velocity of the incident flow is examined in this section. In this case, two more dimensionless parameters number are required to describe the flow in addition to the Reynolds. The time-depended incident flow velocity is given by

$$\mathcal{U}I(t) = \mathcal{U}\_{\infty} + \Delta u \sin\left(4\pi f\_{\mathcal{C}}t\right) \tag{14}$$

where Δ*u* is the amplitude and *fe* is the nominal frequency of the imposed velocity perturbations. It should be pointed out that the actual frequency of the velocity waveform in Eq. (14) is 2 *fe* for consistency with non-harmonic waveforms that will be employed in the next section. In the following, the different cases examined will be described in terms of

$$\text{Reduced velocity} = \frac{\text{U}\_{\infty}}{f\_{\text{c}}D}.$$

$$\text{Amplitude ratio} = \frac{\Delta u}{\text{U}\_{\infty}}.$$

Results are shown below for a constant Reynolds number (*Re* = 150) and a constant amplitude ratio (Δ*u*/*U*∞ = 0.20). The interest here is in the determination of the vortex lock-on range in which the shedding frequency is captured by the perturbation frequency. For this purpose, a number of runs were conducted starting from perturbation frequencies near the vortex shedding frequency in the unperturbed wake for which lock-on is guaranteed, and either increasing or decreasing the perturbation frequency until the wake response is not locked-on.

8 Will-be-set-by-IN-TECH

(a) Lift coefficient (b) Strouhal number

Fig. 4. Flow parameters as a function of Reynolds numbers in steady flow. Symbols show the

Overall, the results presented here for a range of Reynolds numbers in the laminar unsteady wake regime for steady incident flow compare favourably with experimental data and other numerical simulations which provides confidence in the numerical method employed. The simultaneous presentation of the vortex patterns in the wake and fluid forces on the body has

The wake response to harmonic perturbations in the velocity of the incident flow is examined in this section. In this case, two more dimensionless parameters number are required to describe the flow in addition to the Reynolds. The time-depended incident flow velocity is

where Δ*u* is the amplitude and *fe* is the nominal frequency of the imposed velocity perturbations. It should be pointed out that the actual frequency of the velocity waveform in Eq. (14) is 2 *fe* for consistency with non-harmonic waveforms that will be employed in the next section. In the following, the different cases examined will be described in terms of

Reduced velocity = *U*∞

Amplitude ratio = Δ*u*

Results are shown below for a constant Reynolds number (*Re* = 150) and a constant amplitude ratio (Δ*u*/*U*∞ = 0.20). The interest here is in the determination of the vortex lock-on range in which the shedding frequency is captured by the perturbation frequency. For this purpose, a number of runs were conducted starting from perturbation frequencies near the vortex shedding frequency in the unperturbed wake for which lock-on is guaranteed, and either increasing or decreasing the perturbation frequency until the wake response is not

*U*(*t*) = *U*∞ + Δ*u* sin (4*π fet*) (14)

*feD* ,

*U*∞ .

present results while the lines indicate empirical relationships.

revealed effectively the connection between them.

given by

locked-on.

**4. Circular cylinder in harmonically perturbed flow**

Fig. 5. Variation of the Strouhal number *vs.* reduced velocity.

Figure 5 shows the frequency of vortex shedding in terms of the Strouhal number, *St* = *fvsD*/*U*∞, as a function of the excitation frequency in terms of the reduced velocity. If there was no interaction between the imposed perturbations and vortex shedding, the Strouhal number would remain constant at the value indicated by a horizontal dashed line. However, the Strouhal number is affected for all the perturbation frequencies considered. For a range of reduced velocities, vortex shedding locks on to the perturbation frequency as indicated by the solid line. For low reduced velocities, the Strouhal number gets higher than its unperturbed value in the natural wake, and vice versa for high reduced velocities. The two cases marked by open symbols will be examined in more detail below.

Figure 6 shows the instantaneous vorticity distributions for two different reduced velocities together with the variation of the drag and lift forces at three instants over approximately half vortex shedding cycle (cf. Fig. 2). For both cases, single vortices are shed alternatively from each side of the cylinder in a fashion similar to that observed for steady flow. However, the details of vortex formation are different. For example, the peak vorticity values and the size of the vortices is increased in harmonically perturbed flow due to wake resonance. Furthermore, the longitudinal distance between shed vortices varies linearly with the reduced velocity. These changes in the wake patterns are accompanied by considerable changes in the magnitude of the mean and fluctuating forces acting on the cylinder. The unsteady in-line force shows a fifty-fold increase in magnitude whereas the transverse increases by 2.5 times compared to their counterparts in steady flow. The in-line force in unsteady flow contains inertial aerodynamic components due to added mass and pressure gradients, i.e.,

$$F\_M = \frac{1}{4} \rho \pi D^2 C\_M \frac{\text{d}U}{\text{d}t} \tag{15}$$

where *CM* = 1 + *CA* is the inertia coefficient and *CA* the added mass coefficient. Under the conditions examined here, the inertial components of the in-line force dominate over the viscous component. As a consequence, the in-line force is almost in phase with the flow acceleration d*U*/d*t*. On the other hand, the transverse force is not affected by inertial components since there is no perturbation in that direction. In Fig. 6, the waveform of the transverse force deviates markedly from a pure harmonic for both reduced velocities and there is quite a difference between them reflecting the effect of perturbation frequency on *CY*.

*y/D* -4 -2 0 2 4 0

*y/D* -4 -2 0 2 4 0

(b) Transverse rms velocity at *x*/*D* = 3.52

0.2

Fig. 7. Time-averaged flow characteristics for steady and harmonically perturbed incident flow; Solid line: steady flow, dashed line: *U*∞/ *feD* = 5.89, dash-dot line: *U*∞/ *feD* = 5.16. All

For *U*∞/ *feD* = 5.89, the lift attains a very broad maximum which makes it difficult to define precisely the phase, *φL*. In other words, the approximation of the lift force as a harmonic function is very crude. Nevertheless, it can be readily verified that the phase at which *CY*(*t*) is maximum shifts with reduced velocity. E.g., it occurs at *t*/*Tv* ≈ 4.4 for *U*∞/ *feD* = 5.16 and at a slightly earlier time for *U*∞/ *feD* = 5.89. In effect, the perturbation frequency provides a

Different perturbation frequencies not only affect the unsteady flow characteristics but also the time-averaged flow. Figure 7 shows the distributions of mean and rms velocities across the wake for two different reduced velocities in harmonically perturbed flow compared to those in steady flow. Clearly, the distributions are much affected in perturbed flow but also depend on the reduced velocity. When looking at the time-averaged location of flow separation it is observed that *θs* = 119◦ and 117◦ for *U*∞/ *feD* = 5.16 and 5.89, respectively. Hence, the separation point moves forward in perturbed flow compared to its location in steady flow. The variation of the mean drag coefficient over the range of reduced velocities examined for Δ*u*/*U*∞ = 0.20 is shown in Fig. 8. The data show a drag amplification at a reduced velocity of 5.15 reaching a maximum value which is 16% higher than its value for steady flow. Outside

In this section, the effect of modest deviations of the perturbation waveform from a pure harmonic on the wake flow and fluid forces on the cylinder is examined. The time-depended

where *α* is a parameter related to the amplitude of velocity perturbations, *β* sets the mean and median velocity, and the index *n* dictates the perturbation waveform. For *n* = 1, the waveform is pure-tone harmonic as those employed in the previous section. For *n* �= 1, the perturbation waveform comprises an infinite number of harmonics. The effect of different values of *n* has

1 + *α* sin2(2*π fet*)

*n*

(16)

0.4

0.6

Bluff Body Aerodynamics and Wake Control 73

(a) Streamwise mean velocity at *x*/*D* = 1.98

means to control the phasing of vortex shedding from the cylinder.

lock-on response, the drag coefficient is close to that for the unperturbed flow.

*U*(*t*) = *β* ±

**5. Effect of non-harmonic flow perturbations**

incident flow velocity is now given by

cases: *Re* = 150, Δ*u*/*U*∞ = 0.20.

0.2 0.4 0.6 0.8 1 1.2

Fig. 6. Vorticity distributions in the cylinder wake and fluid forces on the cylinder at different instants for two reduced velocities in harmonically perturbed flow; *Re* = 150, Δ*u*/*U*∞ = 0.20.

10 Will-be-set-by-IN-TECH

−1 −0.5 0 0.5 1

4 4.2 4.4 4.6

*t/Tv*

(b) *U*∞/ *feD* = 5.89

*CX*

*CY*

Fig. 6. Vorticity distributions in the cylinder wake and fluid forces on the cylinder at different instants for two reduced velocities in harmonically perturbed flow; *Re* = 150, Δ*u*/*U*∞ = 0.20.

−1 −0.5 0 0.5 1

4 4.2 4.4 4.6

*t/Tv*

(a) *U*∞/ *feD* = 5.16

*CX*

*CY*

(a) Streamwise mean velocity at *x*/*D* = 1.98

Fig. 7. Time-averaged flow characteristics for steady and harmonically perturbed incident flow; Solid line: steady flow, dashed line: *U*∞/ *feD* = 5.89, dash-dot line: *U*∞/ *feD* = 5.16. All cases: *Re* = 150, Δ*u*/*U*∞ = 0.20.

For *U*∞/ *feD* = 5.89, the lift attains a very broad maximum which makes it difficult to define precisely the phase, *φL*. In other words, the approximation of the lift force as a harmonic function is very crude. Nevertheless, it can be readily verified that the phase at which *CY*(*t*) is maximum shifts with reduced velocity. E.g., it occurs at *t*/*Tv* ≈ 4.4 for *U*∞/ *feD* = 5.16 and at a slightly earlier time for *U*∞/ *feD* = 5.89. In effect, the perturbation frequency provides a means to control the phasing of vortex shedding from the cylinder.

Different perturbation frequencies not only affect the unsteady flow characteristics but also the time-averaged flow. Figure 7 shows the distributions of mean and rms velocities across the wake for two different reduced velocities in harmonically perturbed flow compared to those in steady flow. Clearly, the distributions are much affected in perturbed flow but also depend on the reduced velocity. When looking at the time-averaged location of flow separation it is observed that *θs* = 119◦ and 117◦ for *U*∞/ *feD* = 5.16 and 5.89, respectively. Hence, the separation point moves forward in perturbed flow compared to its location in steady flow.

The variation of the mean drag coefficient over the range of reduced velocities examined for Δ*u*/*U*∞ = 0.20 is shown in Fig. 8. The data show a drag amplification at a reduced velocity of 5.15 reaching a maximum value which is 16% higher than its value for steady flow. Outside lock-on response, the drag coefficient is close to that for the unperturbed flow.

#### **5. Effect of non-harmonic flow perturbations**

In this section, the effect of modest deviations of the perturbation waveform from a pure harmonic on the wake flow and fluid forces on the cylinder is examined. The time-depended incident flow velocity is now given by

$$\mathcal{U}(t) = \beta \pm \left[1 + a \sin^2(2\pi f\_\ell t)\right]^n \tag{16}$$

where *α* is a parameter related to the amplitude of velocity perturbations, *β* sets the mean and median velocity, and the index *n* dictates the perturbation waveform. For *n* = 1, the waveform is pure-tone harmonic as those employed in the previous section. For *n* �= 1, the perturbation waveform comprises an infinite number of harmonics. The effect of different values of *n* has

−1 −0.5 0 0.5 1

4 4.2 4.4 4.6

*t/Tv*

−1 −0.5 0 0.5 1

4 4.2 4.4 4.6

*t/Tv*

(b) Minus waveform

*CX*

Bluff Body Aerodynamics and Wake Control 75

*CY*

Fig. 9. Vorticity distributions in the cylinder wake and fluid forces on the cylinder for two different non-harmonic perturbation waveforms; *U*∞/ *feD* = 5.16, Δ*u*/*U*∞ = 0.20, *Re* ≈ 150.

(a) Plus waveform

*CX*

*CY*

Fig. 8. Variation of the mean drag coefficient *vs.* reduced velocity.

been examined previously by the authors (Konstantinidis & Bouris, 2010). For the present study, a value of *n* = −1 was employed and two different waveforms were produced by using different signs in Eq. (16). Therefore, the waveform produced by a plus sign will be denoted 'plus waveform' and likewise for the 'minus waveform'. The alpha parameter was kept constant so that the velocity amplitude Δ*U* = (*U*max − *U*min)/2 = 0.20*U*∞, i.e., the same as that for harmonic perturbations. It should be noted that the time-averaged mean velocity of the incident flow is slightly different for the two different waveforms. Hence, the median velocity *U*0 = (*U*max + *U*min)/2 ≈ *U*∞ is employed as a reference to allow comparisons. Furthermore, due to the large number of independent variables, results will be shown below only for the two perturbation frequencies which have been examined in more detail in the previous section to allow comparisons between different non-harmonic and harmonic waveforms.

Figure 9 shows the instantaneous vorticity distributions for the two different waveforms together with the drag and lift forces at three instants over approximately half vortex shedding cycle at a reduced velocity of 5.16 (cf. Fig. 2(a)). For both waveforms, the wake response remains phase-locked to the imposed perturbation. Comparison of the vorticity contour lines indicates that the details of vortex formation depend on the perturbation waveform. The plus waveform causes the formation of slightly larger vortices than the minus waveform. Furthermore, the vortices appear to be shed slightly earlier for the minus waveform whereas a plus waveform appears to delay vortex shedding. However, it is difficult to quantify these differences. On the other hand, their effect is more evident on the variation of the fluid forces. The in-line force is markedly different for the two waveforms as it follows the flow acceleration d*U*/d*t* due to the dominance of the inertial components. The waveform of the transverse force exhibits some delicate differences, most notably, the time at which it attains its maximum which occurs at a slightly later time for the plus than for the minus waveform. This is consistent with the phasing of vortex shedding above.

Similar observations as those made above concerning the effect of plus and minus waveforms at a reduced velocity of 5.16, also pertain in the case *U*∞/ *feD* = 5.89 for which results are shown in Fig.10.

12 Will-be-set-by-IN-TECH

∞

been examined previously by the authors (Konstantinidis & Bouris, 2010). For the present study, a value of *n* = −1 was employed and two different waveforms were produced by using different signs in Eq. (16). Therefore, the waveform produced by a plus sign will be denoted 'plus waveform' and likewise for the 'minus waveform'. The alpha parameter was kept constant so that the velocity amplitude Δ*U* = (*U*max − *U*min)/2 = 0.20*U*∞, i.e., the same as that for harmonic perturbations. It should be noted that the time-averaged mean velocity of the incident flow is slightly different for the two different waveforms. Hence, the median velocity *U*0 = (*U*max + *U*min)/2 ≈ *U*∞ is employed as a reference to allow comparisons. Furthermore, due to the large number of independent variables, results will be shown below only for the two perturbation frequencies which have been examined in more detail in the previous section to allow comparisons between different non-harmonic

Figure 9 shows the instantaneous vorticity distributions for the two different waveforms together with the drag and lift forces at three instants over approximately half vortex shedding cycle at a reduced velocity of 5.16 (cf. Fig. 2(a)). For both waveforms, the wake response remains phase-locked to the imposed perturbation. Comparison of the vorticity contour lines indicates that the details of vortex formation depend on the perturbation waveform. The plus waveform causes the formation of slightly larger vortices than the minus waveform. Furthermore, the vortices appear to be shed slightly earlier for the minus waveform whereas a plus waveform appears to delay vortex shedding. However, it is difficult to quantify these differences. On the other hand, their effect is more evident on the variation of the fluid forces. The in-line force is markedly different for the two waveforms as it follows the flow acceleration d*U*/d*t* due to the dominance of the inertial components. The waveform of the transverse force exhibits some delicate differences, most notably, the time at which it attains its maximum which occurs at a slightly later time for the plus than for the minus waveform.

Similar observations as those made above concerning the effect of plus and minus waveforms at a reduced velocity of 5.16, also pertain in the case *U*∞/ *feD* = 5.89 for which results are

Fig. 8. Variation of the mean drag coefficient *vs.* reduced velocity.

This is consistent with the phasing of vortex shedding above.

and harmonic waveforms.

shown in Fig.10.

(a) Plus waveform

(b) Minus waveform

Fig. 9. Vorticity distributions in the cylinder wake and fluid forces on the cylinder for two different non-harmonic perturbation waveforms; *U*∞/ *feD* = 5.16, Δ*u*/*U*∞ = 0.20, *Re* ≈ 150.

Plus waveform Minus waveform

5.16 +3.9 +7.1 -3.9 -4.9 5.89 +2.7 +3.0 -2.0 -1.4 Table 2. Percent difference for two different non-harmonic waveforms compared to the fluid

Bluff Body Aerodynamics and Wake Control 77

The differences in the time-averaged wake flow are very small for the two non-harmonic waveforms examined and need not be discussed in detail. On the other hand, the time-averaged mean and fluctuating forces reveal the global effect of the perturbation waveform. Table 2 shows the percent change compared to the harmonically perturbed flow for the same reduced velocity and amplitude ratio. The plus waveform causes an increase in the magnitude of both mean drag and rms lift and the opposite happens for the minus waveform. The change is not equivalent for the plus and minus waveform; it is larger for the plus waveform. In addition, these effects are more pronounced for *U*∞/ *feD* = 5.16 (higher

In this chapter, issues related to the aerodynamics of bluff bodies in steady and time-dependent flows were presented. The subject was treated by numerical simulations of the 2D fluid flow around a circular cylinder which provided information on the dynamics of vortex formation and shedding in the wake and the induced forces acting on the body. Simulations were conducted for low Reynolds numbers in the laminar wake regime, mostly at *Re* ≈ 150. Although this approach holds simplifying assumptions compared to real flows of practical interest, the observed phenomena convey over a wide range of Reynolds numbers and, in fact, makes it easier to discern the effects which are otherwise masked by wake turbulence and shear-layer instabilities that develop at higher Reynolds numbers. The results demonstrate that the vortex formation and the wake flow can be controlled by perturbations in the incident flow velocity. This control can be exercised by appropriately selecting the frequency and amplitude of the imposed perturbations on both the mean and fluctuating wake flow and, thereby, on the fluid forces on the cylinder. Whereas harmonic perturbations have been often utilized in the past, the present study also addresses the effect of non-harmonic perturbations, i.e. perturbations whose waveform deviates from a pure-tone harmonic. The findings of the present study illustrate that more effective wake control can be achieved by adjusting the waveform in addition to the perturbation frequency and amplitude.

Bearman, P. (1997). Near wake flows behind two- and three-dimensional bluff bodies, *Journal*

Gerrard, J. H. (1966). The three-dimensional structure of the wake of a circular cylinder, *Journal*

Griffin, O. M. & Hall, M. S. (1991). Vortex shedding lock-on and flow control in bluff body wakes – Review, *Journal of Fluids Engineering* Vol. 113(No. 4): 526–537. Henderson, R. (1995). Details of the drag curve near the onset of vortex shedding, *Physics of*

*of Wind Engineering and Industrial Aerodynamics* Vol. 71: 33–54.

*of Fluid Mechanics* Vol. 25(Part 1): 143–164.

*Fluids* Vol. 7(No. 9): 2102–2104.

*L CD C*�

*L*

*U*∞/ *feD CD C*�

forces calculated for harmonically perturbed flow.

perturbation frequency) than for *U*∞/ *feD* = 5.89.

**6. Conclusion**

**7. References**

(a) Plus waveform

(b) Minus waveform

Fig. 10. Vorticity distributions in the cylinder wake and fluid forces on the cylinder for two different non-harmonic perturbation waveforms; *U*∞/ *feD* = 5.89, Δ*u*/*U*∞ = 0.20, *Re* ≈ 150.

Table 2. Percent difference for two different non-harmonic waveforms compared to the fluid forces calculated for harmonically perturbed flow.

The differences in the time-averaged wake flow are very small for the two non-harmonic waveforms examined and need not be discussed in detail. On the other hand, the time-averaged mean and fluctuating forces reveal the global effect of the perturbation waveform. Table 2 shows the percent change compared to the harmonically perturbed flow for the same reduced velocity and amplitude ratio. The plus waveform causes an increase in the magnitude of both mean drag and rms lift and the opposite happens for the minus waveform. The change is not equivalent for the plus and minus waveform; it is larger for the plus waveform. In addition, these effects are more pronounced for *U*∞/ *feD* = 5.16 (higher perturbation frequency) than for *U*∞/ *feD* = 5.89.

#### **6. Conclusion**

14 Will-be-set-by-IN-TECH

−1 −0.5 0 0.5 1

4 4.2 4.4 4.6

*t/Tv*

(b) Minus waveform

*CX*

*CY*

Fig. 10. Vorticity distributions in the cylinder wake and fluid forces on the cylinder for two different non-harmonic perturbation waveforms; *U*∞/ *feD* = 5.89, Δ*u*/*U*∞ = 0.20, *Re* ≈ 150.

−1 −0.5 0 0.5 1

4 4.2 4.4 4.6

*t/Tv*

(a) Plus waveform

*CX*

*CY*

In this chapter, issues related to the aerodynamics of bluff bodies in steady and time-dependent flows were presented. The subject was treated by numerical simulations of the 2D fluid flow around a circular cylinder which provided information on the dynamics of vortex formation and shedding in the wake and the induced forces acting on the body. Simulations were conducted for low Reynolds numbers in the laminar wake regime, mostly at *Re* ≈ 150. Although this approach holds simplifying assumptions compared to real flows of practical interest, the observed phenomena convey over a wide range of Reynolds numbers and, in fact, makes it easier to discern the effects which are otherwise masked by wake turbulence and shear-layer instabilities that develop at higher Reynolds numbers. The results demonstrate that the vortex formation and the wake flow can be controlled by perturbations in the incident flow velocity. This control can be exercised by appropriately selecting the frequency and amplitude of the imposed perturbations on both the mean and fluctuating wake flow and, thereby, on the fluid forces on the cylinder. Whereas harmonic perturbations have been often utilized in the past, the present study also addresses the effect of non-harmonic perturbations, i.e. perturbations whose waveform deviates from a pure-tone harmonic. The findings of the present study illustrate that more effective wake control can be achieved by adjusting the waveform in addition to the perturbation frequency and amplitude.

#### **7. References**

**0**

**5**

*France*

**Simulation of Flow Control with Microjets for**

With the increasing flight traffic in the past years grew up several types of pollutions for all near airports inhabitants. One of those major pollutions is the noise generated by aircrafts at takeoff. Despite all the work performed during the last decades on the understanding of the noise generation and its reduction, this nuisance still remains a really challenging problem. Above all the noise sources on a plane during takeoff, jet noise remains the dominant one,

For subsonic jets, radiated noise is a consequence of flow mixing. It is now universally admitted that this mixing noise is generated by the turbulence of the flow and that the noise producing region is axially restricted to about two potential core lengths (Fisher et al., 1977; Laufer et al., 1976). Turbulence is separated in this region in small turbulent eddies, with small dimensions compared to the nozzle, and large-scale structures. It is especially confirmed by the experimental observations (Tam et al., 2008) that both fine- and large-scale structures are the noise sources, large turbulent eddies being predominant in downstream direction

Based on these observations, noise reduction devices are designed to act on jet turbulence development and especially to decrease the growth of the large scale structures in order to reduce major noise generation mechanisms. Several passive and active processes such as chevrons (Bridges & Brown, 2004; Nesbitt & Young, 2008) and microjets are currently being investigated experimentally and numerically. A major advantage of microjets, compared to passive devices, is their possibility to be turned off during cruise, which limits the thrust loss

The use of continuous air microjets on Mach 0.9 high Reynolds round jets has been experimentally investigated in the recent years (Alkislar et al., 2007; Arakeri et al., 2003; Castelain, 2006; Castelain et al., 2007; 2008). Measured data show that the interaction between each actuator and the jet shear layer corrugates the main flow over a distance of two to three jet diameters after the nozzle exit and generates a pair of counter-rotating axial vortices downstream of each microjet. All authors also observed a reduction of turbulence intensities in the shear layer and a lengthening of the potential core, with the exception of Alkislar et al. who noticed a turbulence increase 2 jet diameters after the nozzle exit, before its reduction downstream. In the far field, sound is decreased by 0.5 dB to 2 dB for all observation angles, which corresponds to low and medium frequency spectra reductions originating from the collapse of large turbulent structures. Higher frequency levels are increased by the microjets; this high frequency lift is explained by the enhancing of the small-scale structures by the fluid

**1. Introduction**

especially for high speed jets.

to takeoff configuration only, for instance.

justifying all the efforts still performed on the topic.

**Subsonic Jet Noise Reduction**

Maxime Huet, Gilles Rahier and François Vuillot *Onera – The French Aerospace Lab, F-92322 Châtillon*

## **Simulation of Flow Control with Microjets for Subsonic Jet Noise Reduction**

Maxime Huet, Gilles Rahier and François Vuillot *Onera – The French Aerospace Lab, F-92322 Châtillon France*

#### **1. Introduction**

16 Will-be-set-by-IN-TECH

78 Applied Aerodynamics

Konstantinidis, E. & Balabani, S. (2007). Symmetric vortex shedding in the near wake of a

Konstantinidis, E. & Balabani, S. (2008). Flow structure in the locked-on wake of a circular

Konstantinidis, E. & Bouris, D. (2010). The effect of nonharmonic forcing on bluff-body

Norberg, C. (2003). Fluctuating lift on a circular cylinder: review and new measurements,

Papadakis, G. & Bergeles, G. (1995). A locally modified 2nd-order upwind scheme for

Rhie, C. M. & Chow, W. L. (1983). Numerical study of the turbulent - Flow past an airfoil with

Triantafyllou, G., Kupfer, K. & Bers, A. (1987). Absolute instabilities and self-sustained

Wu, M. H., Wen, C. Y., Yen, R. H., Weng, M. C. & Wang, A. B. (2004). Experimental and

trailing edge separation, *AIAA Journal* Vol. 21( No. 11): 1525–1532.

Reynolds number, *Journal of Fluid Mechanics* Vol. 515: 233–260.

23: 1047–1063.

17): 1914–1917.

*and Fluid Flow* Vol. 29(No. 6): 1567–1576.

*Journal of Fluids and Structures* Vol. 17( No. 1): 57–96.

*Aerodynamics* Vol. 98: 245–252.

*Fluid Flow* Vol. 5( No. 1): 49–62.

circular cylinder due to streamwise perturbations, *Journal of Fluids and Structures* Vol.

cylinder in pulsating flow: Effect of forcing amplitude, *International Journal of Heat*

aerodynamics at a low Reynolds number, *Jouranl of Wind Engineering and Industrial*

convection terms discretization, *International Journal of Numerical Methods for Heat and*

oscillations in the wakes of circular-cylinders, *Physical Review Letters* Vol. 59(No.

numerical study of the separation angle for flow around a circular cylinder at low

With the increasing flight traffic in the past years grew up several types of pollutions for all near airports inhabitants. One of those major pollutions is the noise generated by aircrafts at takeoff. Despite all the work performed during the last decades on the understanding of the noise generation and its reduction, this nuisance still remains a really challenging problem. Above all the noise sources on a plane during takeoff, jet noise remains the dominant one, justifying all the efforts still performed on the topic.

For subsonic jets, radiated noise is a consequence of flow mixing. It is now universally admitted that this mixing noise is generated by the turbulence of the flow and that the noise producing region is axially restricted to about two potential core lengths (Fisher et al., 1977; Laufer et al., 1976). Turbulence is separated in this region in small turbulent eddies, with small dimensions compared to the nozzle, and large-scale structures. It is especially confirmed by the experimental observations (Tam et al., 2008) that both fine- and large-scale structures are the noise sources, large turbulent eddies being predominant in downstream direction especially for high speed jets.

Based on these observations, noise reduction devices are designed to act on jet turbulence development and especially to decrease the growth of the large scale structures in order to reduce major noise generation mechanisms. Several passive and active processes such as chevrons (Bridges & Brown, 2004; Nesbitt & Young, 2008) and microjets are currently being investigated experimentally and numerically. A major advantage of microjets, compared to passive devices, is their possibility to be turned off during cruise, which limits the thrust loss to takeoff configuration only, for instance.

The use of continuous air microjets on Mach 0.9 high Reynolds round jets has been experimentally investigated in the recent years (Alkislar et al., 2007; Arakeri et al., 2003; Castelain, 2006; Castelain et al., 2007; 2008). Measured data show that the interaction between each actuator and the jet shear layer corrugates the main flow over a distance of two to three jet diameters after the nozzle exit and generates a pair of counter-rotating axial vortices downstream of each microjet. All authors also observed a reduction of turbulence intensities in the shear layer and a lengthening of the potential core, with the exception of Alkislar et al. who noticed a turbulence increase 2 jet diameters after the nozzle exit, before its reduction downstream. In the far field, sound is decreased by 0.5 dB to 2 dB for all observation angles, which corresponds to low and medium frequency spectra reductions originating from the collapse of large turbulent structures. Higher frequency levels are increased by the microjets; this high frequency lift is explained by the enhancing of the small-scale structures by the fluid

analysis. Acoustic results are then reproduced in section 4. Comparisons between simulations and experiments are made when available and changes in spectra shapes and integrated levels are analyzed. Relations are made with the evolutions previously noticed in the flow. Paper

Simulation of Flow Control with Microjets for Subsonic Jet Noise Reduction 81

Aeroacoustics simulations are performed in two separate steps. Instantaneous flow simulations are conducted in a first time to compute the noise sources in the jet. To provide the best possible flow resolution with existing computed resources, accurate Navier-Stokes simulations are restricted to the jet plume, during which instantaneous aerodynamic fields are stored on surfaces surrounding the jet. Those fields are then used as source data for noise

Aerodynamic simulations are performed using the flow solver CEDRE developed at Onera. CEDRE is a multi-physics, reactive solver used by researchers and aeronautical industries for engine conception and optimisation, such as combustion (Dorey et al., 2010; Dupoirieux & Bertier, 2011), turbine blade cooling (Guillou & Chedevergne, 2011) and jet noise (Bodard et al., 2009), for instance. Navier-Stokes equations are solved using second order upwind schemes space discretization for generalised polyhedral computational grids (Courbet et al., 2011) with explicit or implicit time schemes from first to third order for time resolution.

In the present simulations, the fluctuating jet flow is solved using the LES model with the MILES approach (Boris et al., 1992; Fureby & Grinstein, 1999; Grinstein & Fureby, 2002), where the dissipation of the structures smaller than the grid size is not modelled and is assumed to be of the order of the numerical dissipation. The validity of this approach for jet simulations relies on the hypothesis proposed by Biancherin (Biancherin, 2003) that those unresolved scales do not notably influence the noise generation in the flow. This assumption has been verified by Muller (Muller et al., 2006), who compared jet flow and noise results obtained using either the Smagorinsky subgrid scale model (Smagorinsky, 1963) or the MILES approach and observed

The current approach developed for jet noise simulations is based on the inclusion of the nozzle geometry in the computational domain, which provides many advantages compared

The first advantage is the absence of artificial disturbance at nozzle exit to destabilize the flow. Freund (Freund, 2001) and Bogey (Bogey et al., 2003), for instance, illustrated the necessity to add numerical perturbations over the imposed mean flow profile for the jet to be destabilized and to generate turbulence, with the risk of generating spurious noise. With the present approach, it is expected that by removing boundary conditions from a critical location, where instability waves must be allowed to develop freely, one permits some kind of natural growth of instability waves to occur. Indeed, Biancherin (Biancherin, 2003) and Muller (Muller, 2006) highlighted that, with this procedure, small truncation errors destabilize the flow and no more

The inclusion of the nozzle all the more permits the simulation of complex, realistic geometries such as short-cowl nozzles with the inclusion of a pylon, bifurcations and chevrons (Eschricht

finally ends with concluding remarks and perspectives in section 5.

radiation to the microphones, using a surface integral method.

**2.1 Aerodynamic numerical specifications**

to simulations limited to flow domain only.

artificial excitation is needed at nozzle exit.

**2. Simulation parameters**

very similar results.

injection. Such turbulence modifications and related noise variations are similar to what is observed with chevrons, for instance (Alkislar et al., 2007).

A more important noise reduction up to 6 dB has been observed using water injection (Krothapalli et al., 2003), which did not exhibit high frequency lift. From these observations, Zaman (Zaman, 2010) suggests that this high frequency crossover is at least partly due to microjets self noise. Water injection nevertheless remains difficult to use on airplanes because of the important weight increase caused by the required water storage.

In addition to the above-mentioned actuators, fluidic control has also been investigated using pulsed microjets. When dealing with fluid injection, a major interest of such a control is the reduction of the injected mass flow rate. Ragaller (Ragaller et al., 2009) for instance demonstrated the capacity of achieving with a reduced mass flow pulsed control a noise decrease close to that obtained with continuous water injection.

Noise reduction through energy injection using plasma actuators is also being investigated for subsonic (Kastner et al., 2008; Kearney-Fischer et al., 2009a;b; Kim et al., 2009; Samimy et al., 2007a) and supersonic (Kearney-Fischer et al., 2011; Samimy et al., 2007b) jets. Control is made with localized arc filament plasma actuators, each actuator consisting in a pair of pin electrodes and generating electric discharge plasmas at a driven frequency varying from 0 to 200 kHz. Authors especially observed a broadband noise increase for low forcing frequency excitation, below *StF* = 1, and a reduction for higher frequencies. This noise reduction is especially improved with increasing main jet temperature.

Only a limited number of numerical works have been published on the action of continuous air microjets for noise reduction (Enomoto et al., 2011; Laurendeau et al., 2008; Najafi-Yazdi et al., 2011; Rife & Page, 2011; Shur et al., 2011). Except for the simulation of Lew (Lew et al., 2010), they all modelled the microjets to avoid gridding the feed pipes. Those simulations illustrate the capacity of Large Eddy Simulations (LES) to capture the effect of microjets on both flow and noise, even using modelled actuators. To the knowledge of the authors, no simulations have been published on the action of pulsed actuators for jet noise control.

In the present work, a circular jet with an acoustic Mach number *Ma* = *Uj*/*c*0 = 0.9 (*Uj* being the axial jet exhaust velocity and *c*0 the ambient sound speed) is computed by LES. Simulations are performed for two main jet temperatures, corresponding to isothermal (*Tj* = 288*K*) and heated (*Tj* = 576*K*) flows with a Reynolds number based on nozzle diameter *D* of *ReD* = *UjD*/*ν* = 1, 000, 000 and 320,000 respectively, where *ν* stands for the kinematic viscosity.

The choice of those configurations is justified by the presence of aerodynamic and acoustic measurements previously performed without control by Institut PPRIME in Poitiers, France, during the JEAN European project, as well as aerodynamic and acoustic measurements realised by Castelain for the same nozzle and isothermal jet condition, with similar microjets configurations (Castelain et al., 2007; 2008). To complete those data, a new set of acoustic measurements has been conducted by Institut PPRIME during the project, to provide far field pressure data for the baseline and continuous microjets simulated configurations.

The paper is organised as follows. The test cases used for the simulations are recalled in section 2, with details given on the methodology used for both aerodynamic and acoustic simulations. Special focus is made on the modelling of the microjets. Section 3 corresponds to a detailed analysis of the aerodynamic fields. Mean and r.m.s. velocity profiles are presented and flow modifications with control are discussed. This part ends with a linear jet stability analysis. Acoustic results are then reproduced in section 4. Comparisons between simulations and experiments are made when available and changes in spectra shapes and integrated levels are analyzed. Relations are made with the evolutions previously noticed in the flow. Paper finally ends with concluding remarks and perspectives in section 5.

#### **2. Simulation parameters**

2 Will-be-set-by-IN-TECH

injection. Such turbulence modifications and related noise variations are similar to what is

A more important noise reduction up to 6 dB has been observed using water injection (Krothapalli et al., 2003), which did not exhibit high frequency lift. From these observations, Zaman (Zaman, 2010) suggests that this high frequency crossover is at least partly due to microjets self noise. Water injection nevertheless remains difficult to use on airplanes because

In addition to the above-mentioned actuators, fluidic control has also been investigated using pulsed microjets. When dealing with fluid injection, a major interest of such a control is the reduction of the injected mass flow rate. Ragaller (Ragaller et al., 2009) for instance demonstrated the capacity of achieving with a reduced mass flow pulsed control a noise

Noise reduction through energy injection using plasma actuators is also being investigated for subsonic (Kastner et al., 2008; Kearney-Fischer et al., 2009a;b; Kim et al., 2009; Samimy et al., 2007a) and supersonic (Kearney-Fischer et al., 2011; Samimy et al., 2007b) jets. Control is made with localized arc filament plasma actuators, each actuator consisting in a pair of pin electrodes and generating electric discharge plasmas at a driven frequency varying from 0 to 200 kHz. Authors especially observed a broadband noise increase for low forcing frequency excitation, below *StF* = 1, and a reduction for higher frequencies. This noise reduction is

Only a limited number of numerical works have been published on the action of continuous air microjets for noise reduction (Enomoto et al., 2011; Laurendeau et al., 2008; Najafi-Yazdi et al., 2011; Rife & Page, 2011; Shur et al., 2011). Except for the simulation of Lew (Lew et al., 2010), they all modelled the microjets to avoid gridding the feed pipes. Those simulations illustrate the capacity of Large Eddy Simulations (LES) to capture the effect of microjets on both flow and noise, even using modelled actuators. To the knowledge of the authors, no simulations have been published on the action of pulsed actuators for jet noise control.

In the present work, a circular jet with an acoustic Mach number *Ma* = *Uj*/*c*0 = 0.9 (*Uj* being the axial jet exhaust velocity and *c*0 the ambient sound speed) is computed by LES. Simulations are performed for two main jet temperatures, corresponding to isothermal (*Tj* = 288*K*) and heated (*Tj* = 576*K*) flows with a Reynolds number based on nozzle diameter *D* of *ReD* = *UjD*/*ν* = 1, 000, 000 and 320,000 respectively, where *ν* stands for the kinematic

The choice of those configurations is justified by the presence of aerodynamic and acoustic measurements previously performed without control by Institut PPRIME in Poitiers, France, during the JEAN European project, as well as aerodynamic and acoustic measurements realised by Castelain for the same nozzle and isothermal jet condition, with similar microjets configurations (Castelain et al., 2007; 2008). To complete those data, a new set of acoustic measurements has been conducted by Institut PPRIME during the project, to provide far field

The paper is organised as follows. The test cases used for the simulations are recalled in section 2, with details given on the methodology used for both aerodynamic and acoustic simulations. Special focus is made on the modelling of the microjets. Section 3 corresponds to a detailed analysis of the aerodynamic fields. Mean and r.m.s. velocity profiles are presented and flow modifications with control are discussed. This part ends with a linear jet stability

pressure data for the baseline and continuous microjets simulated configurations.

observed with chevrons, for instance (Alkislar et al., 2007).

of the important weight increase caused by the required water storage.

decrease close to that obtained with continuous water injection.

especially improved with increasing main jet temperature.

viscosity.

Aeroacoustics simulations are performed in two separate steps. Instantaneous flow simulations are conducted in a first time to compute the noise sources in the jet. To provide the best possible flow resolution with existing computed resources, accurate Navier-Stokes simulations are restricted to the jet plume, during which instantaneous aerodynamic fields are stored on surfaces surrounding the jet. Those fields are then used as source data for noise radiation to the microphones, using a surface integral method.

#### **2.1 Aerodynamic numerical specifications**

Aerodynamic simulations are performed using the flow solver CEDRE developed at Onera. CEDRE is a multi-physics, reactive solver used by researchers and aeronautical industries for engine conception and optimisation, such as combustion (Dorey et al., 2010; Dupoirieux & Bertier, 2011), turbine blade cooling (Guillou & Chedevergne, 2011) and jet noise (Bodard et al., 2009), for instance. Navier-Stokes equations are solved using second order upwind schemes space discretization for generalised polyhedral computational grids (Courbet et al., 2011) with explicit or implicit time schemes from first to third order for time resolution.

In the present simulations, the fluctuating jet flow is solved using the LES model with the MILES approach (Boris et al., 1992; Fureby & Grinstein, 1999; Grinstein & Fureby, 2002), where the dissipation of the structures smaller than the grid size is not modelled and is assumed to be of the order of the numerical dissipation. The validity of this approach for jet simulations relies on the hypothesis proposed by Biancherin (Biancherin, 2003) that those unresolved scales do not notably influence the noise generation in the flow. This assumption has been verified by Muller (Muller et al., 2006), who compared jet flow and noise results obtained using either the Smagorinsky subgrid scale model (Smagorinsky, 1963) or the MILES approach and observed very similar results.

The current approach developed for jet noise simulations is based on the inclusion of the nozzle geometry in the computational domain, which provides many advantages compared to simulations limited to flow domain only.

The first advantage is the absence of artificial disturbance at nozzle exit to destabilize the flow. Freund (Freund, 2001) and Bogey (Bogey et al., 2003), for instance, illustrated the necessity to add numerical perturbations over the imposed mean flow profile for the jet to be destabilized and to generate turbulence, with the risk of generating spurious noise. With the present approach, it is expected that by removing boundary conditions from a critical location, where instability waves must be allowed to develop freely, one permits some kind of natural growth of instability waves to occur. Indeed, Biancherin (Biancherin, 2003) and Muller (Muller, 2006) highlighted that, with this procedure, small truncation errors destabilize the flow and no more artificial excitation is needed at nozzle exit.

The inclusion of the nozzle all the more permits the simulation of complex, realistic geometries such as short-cowl nozzles with the inclusion of a pylon, bifurcations and chevrons (Eschricht

Fig. 2. Detail of the O-grid configuration used on jet axis. Axial position *x*/*D* = 1.

*D* = 0.153 *x*

*δ*

especially after the end of the potential core.

An illustration of the mesh at nozzle exit is given on Fig. 3 (a).

Muller, 2006):

shear layer thickness *δ* by Candel and already used in previous simulations (Biancherin, 2003;

Simulation of Flow Control with Microjets for Subsonic Jet Noise Reduction 83

*D*

Outside of the flow development region, an acoustic criterion is chosen to properly propagate acoustic waves to the storage surfaces used for noise radiation (see §2.3). For subsonic jets, where noise comes from flow mixing, dominant pressure waves are observed for Strouhal numbers *St* = *f D*/*Uj* below 0.3. It is thus chosen to radiate noise with negligible dissipation up to *St* = 0.5, which provides most of the energy contained in the pressure spectra. Biancherin (Biancherin, 2003) demonstrated that, with the numerical methods used in the flow solver, a minimum of 20 points per wavelength is required for a correct propagation of the pressures waves over the distances considered with the constructed numerical grid.

In the present case, radial resolution is set to resolve the shear layer with 20 cells at an axial distance of 1 *D* after the nozzle exit, where the radial size of the cells is chosen to double between centreline and exterior of the shear with a regular growth of 7%. The same radial growth rate is used outside of the shear layer up to the end of the structured patch and satisfies the acoustic criterion for correct noise resolution. A 9.5% growth rate is used between the shear layer and the jet axis, where larger turbulent structures are expected to be observed

Axial size of the elements is imposed at nozzle exit and is similar to the radial extent of the elements used in the shear layer at jet exhaust. Axial stretching of 1% is used until the elements reach the maximum size given by the acoustical criterion, after what no growth rate is applied. The structured patch is composed of 1.6 × 106 hexahedra, for a total mesh of 4.5 × 106 cells.

Construction of such grids leads to a limited amount of very small elements close to the nozzle exit, where velocity and temperature reach their maximum values. The use of an explicit time scheme would thus oblige one to run simulations with a very low time step verifying the CFL condition (*u* + *c*) Δ*t*/Δ*x* < 1 , which can be very penalizing for efficient and quick simulations. Proper flow resolution in those elements not being critical for an accurate description of the whole jet, time resolution is performed with a first order implicit time scheme that allows CFL criterion above 1 in those cells, and thus an acceptable time step. Use of such a scheme is not a problem for noise propagation: results from previous

+ 0.002 (1)

Fig. 1. Illustration of half the computational domain. Orange elements correspond to external boundaries, yellow to nozzle surfaces, blue to upstream fairing, red to structured patch and black to unstructured elements.

et al., 2008; Uzun & Hussaini, 2009; Xia et al., 2008; Yan et al., 2007). The increased difficulty in mesh generation is overtaken in the present case by the use of an unstructured grid generator.

The full numerical domain is a cylinder with length and radius of 120 *D* and 84 *D* respectively, mainly composed of tetrahedrons. Those important dimensions, far greater than the requested noise source generation domain, give possible the use of large sponge zones to avoid spurious noise reflections at the outlet boundaries of the computational domain, where static pressure is imposed. Inlet boundaries correspond to imposed total pressure and temperature at nozzle inlet and imposed velocity and static temperature at external flow inlet.

According to Zaman (Zaman, 1985), nozzle exit boundary-layer momentum thickness of a high Reynolds number jet of about 106 is *δθ*/*D* ∼ 10−3. Proper numerical resolution of such a thin boundary layer is unreachable with present computational power. The objective of the study being to illustrate the capacity of the simulations to quantitatively reproduce the noise reduction effect of microjets for industrial interests, it is chosen to restrict the grid size to affordable simulations with preferred high resolution of the flow in noise production regions. Boundary layers are thus under-resolved in the present computations, with the momentum thickness being roughly *δθ*/*D* ∼ 0.2. This value is for instance four times larger than the one already used for other simulations of Mach 0.9 jets (Bodony & Lele, 2005; Bogey et al., 2003).

To increase resolution of the turbulence in the jet plume, a structured patch represented in red on Fig. 1 is included in the computational grid behind the nozzle exit. This truncated cone extends axially from *x*/*D* = 0 to *x*/*D* = 25, and radially to *r*/*D* = 2 and 4, respectively, at the two previous axial positions. Its construction is based on a 2D grid, rotated around the jet axis to provide 60 azimuthal hexahedral elements. The central part of the mesh is modified with a O-grid configuration illustrated on Fig. 2 to ensure homogeneous sized cells in the jet core.

The construction of the structured patch is based on aerodynamic and acoustic criteria to ensure sufficient resolution of flow development and sound propagation. First criterion ensures a sufficient resolution of the sheared flow and relies on the estimation of the jet 4 Will-be-set-by-IN-TECH

Fig. 1. Illustration of half the computational domain. Orange elements correspond to external boundaries, yellow to nozzle surfaces, blue to upstream fairing, red to structured patch and

et al., 2008; Uzun & Hussaini, 2009; Xia et al., 2008; Yan et al., 2007). The increased difficulty in mesh generation is overtaken in the present case by the use of an unstructured grid generator. The full numerical domain is a cylinder with length and radius of 120 *D* and 84 *D* respectively, mainly composed of tetrahedrons. Those important dimensions, far greater than the requested noise source generation domain, give possible the use of large sponge zones to avoid spurious noise reflections at the outlet boundaries of the computational domain, where static pressure is imposed. Inlet boundaries correspond to imposed total pressure and temperature at nozzle inlet and imposed velocity and static temperature at external flow inlet. According to Zaman (Zaman, 1985), nozzle exit boundary-layer momentum thickness of a high Reynolds number jet of about 106 is *δθ*/*D* ∼ 10−3. Proper numerical resolution of such a thin boundary layer is unreachable with present computational power. The objective of the study being to illustrate the capacity of the simulations to quantitatively reproduce the noise reduction effect of microjets for industrial interests, it is chosen to restrict the grid size to affordable simulations with preferred high resolution of the flow in noise production regions. Boundary layers are thus under-resolved in the present computations, with the momentum thickness being roughly *δθ*/*D* ∼ 0.2. This value is for instance four times larger than the one already used for other simulations of Mach 0.9 jets (Bodony & Lele, 2005; Bogey et al., 2003). To increase resolution of the turbulence in the jet plume, a structured patch represented in red on Fig. 1 is included in the computational grid behind the nozzle exit. This truncated cone extends axially from *x*/*D* = 0 to *x*/*D* = 25, and radially to *r*/*D* = 2 and 4, respectively, at the two previous axial positions. Its construction is based on a 2D grid, rotated around the jet axis to provide 60 azimuthal hexahedral elements. The central part of the mesh is modified with a O-grid configuration illustrated on Fig. 2 to ensure homogeneous sized cells in the jet core. The construction of the structured patch is based on aerodynamic and acoustic criteria to ensure sufficient resolution of flow development and sound propagation. First criterion ensures a sufficient resolution of the sheared flow and relies on the estimation of the jet

black to unstructured elements.

Fig. 2. Detail of the O-grid configuration used on jet axis. Axial position *x*/*D* = 1.

shear layer thickness *δ* by Candel and already used in previous simulations (Biancherin, 2003; Muller, 2006):

$$\frac{\delta}{D} = 0.153 \left( \frac{\chi}{D} \right) + 0.002 \tag{1}$$

Outside of the flow development region, an acoustic criterion is chosen to properly propagate acoustic waves to the storage surfaces used for noise radiation (see §2.3). For subsonic jets, where noise comes from flow mixing, dominant pressure waves are observed for Strouhal numbers *St* = *f D*/*Uj* below 0.3. It is thus chosen to radiate noise with negligible dissipation up to *St* = 0.5, which provides most of the energy contained in the pressure spectra. Biancherin (Biancherin, 2003) demonstrated that, with the numerical methods used in the flow solver, a minimum of 20 points per wavelength is required for a correct propagation of the pressures waves over the distances considered with the constructed numerical grid.

In the present case, radial resolution is set to resolve the shear layer with 20 cells at an axial distance of 1 *D* after the nozzle exit, where the radial size of the cells is chosen to double between centreline and exterior of the shear with a regular growth of 7%. The same radial growth rate is used outside of the shear layer up to the end of the structured patch and satisfies the acoustic criterion for correct noise resolution. A 9.5% growth rate is used between the shear layer and the jet axis, where larger turbulent structures are expected to be observed especially after the end of the potential core.

Axial size of the elements is imposed at nozzle exit and is similar to the radial extent of the elements used in the shear layer at jet exhaust. Axial stretching of 1% is used until the elements reach the maximum size given by the acoustical criterion, after what no growth rate is applied.

The structured patch is composed of 1.6 × 106 hexahedra, for a total mesh of 4.5 × 106 cells. An illustration of the mesh at nozzle exit is given on Fig. 3 (a).

Construction of such grids leads to a limited amount of very small elements close to the nozzle exit, where velocity and temperature reach their maximum values. The use of an explicit time scheme would thus oblige one to run simulations with a very low time step verifying the CFL condition (*u* + *c*) Δ*t*/Δ*x* < 1 , which can be very penalizing for efficient and quick simulations. Proper flow resolution in those elements not being critical for an accurate description of the whole jet, time resolution is performed with a first order implicit time scheme that allows CFL criterion above 1 in those cells, and thus an acceptable time step. Use of such a scheme is not a problem for noise propagation: results from previous

To ensure that the computed velocity and temperature correspond to the desired values, the mass flow rate *m*˙ is determined through an iterative process. Based on the aimed velocity and temperature, an estimation of the initial mass flow rate can be calculated using the fluid velocity, its density and the surface *S* used for the injection: *m*˙ = *ρUS*, where density is approximated by considering that the microjet flow occurs at ambient pressure. The surface

Simulation of Flow Control with Microjets for Subsonic Jet Noise Reduction 85

sum of downstream exhaust surfaces *Si* of the hexahedral cell used for microjet injection and

Using this iterative approach, mass flow rate is updated at each iteration until velocity and temperature reach the target values. From the computed surface used for the injection, it is in addition possible to evaluate the equivalent microjet diameter, considering a circular microjet

Instantaneous flow fields provided by the flow simulation and stored on surfaces are used to radiate sound to the microphones. Several integral formulations such as Lighthill (Lighthill, 1952), Kirchhoff (Lyrintzis, 1994) and Ffowcs Williams & Hawkings (FW-H) (Ffowcs Williams & Hawkings, 1969) are available to reconstruct far field noise from volume or surface fluctuations. It is usually preferred to use surface fields to limit the amount of stored data and numerical operations required to perform the noise radiation, which is done in the present study. Lighthill volume integral formulation can nevertheless be fruitful if one wants to identify the noise sources present in the flow (in Lighthill's sense), as done by Perez (Perez

Based on the extended investigations performed on the topic by Rahier (Rahier et al., 2004), noise radiation is performed in the present case using FW-H porous surface formulation. In their paper, the authors especially demonstrated that this formulation makes it possible to place the control surface rather close to the jet, in a non-uniform flow. The surface position requirement is that the non-uniformities of the flow on the control surface are only due to convected gradients (such as density gradients related to thermal mixing in the case of a hot jet) and are not high sources of noise production (such as instability expanding or vortex pairing). On the contrary, the Kirchhoff method (with density or pressure as input data) is a much less suited tool for acoustic post-processing of jet aerodynamic simulations because of its high sensibility to non acoustic density or pressure gradients (local thermal mixing or

Control surfaces are moreover kept open at both extremities. Investigations indeed showed that closing the surface leads in the best case to results similar as with open surface and can provide erroneous results if the control surface is too short, because of the turbulence level on the downstream closing disc. As a consequence of using an open control surface, its axial extent needs to be sufficiently large for a correct radiation at low angles, where the maximum

An axial extent of 25 diameters is used for the present simulations. Surfaces additionally verify 1 ≤ *r*/*D* ≤ 2 at nozzle exit and 3 ≤ *r*/*D* ≤ 4 at the downstream end. Radiation is performed for 4 different surfaces. Collapsing time signals and pressure spectra in the far field ensure that the surfaces enclose all noise sources and provide a limited dissipation for

noise is expected. It practically leads to a length of 20 to 25 nozzle diameters.

frequencies below a Strouhal number of 0.5.

*S*/*U* with

*S* = ∑*i Si* ·*ni* the

can be evaluated from cell geometry and velocity vector: *S* = *U* ·

*ni* their associated outside-oriented normal vectors.

√*S*/*π*.

nozzle : *deq* = 2

et al., 2007).

vorticity).

**2.3 Noise radiation**

(a) detail of nozzle exit and mesh adaptation to model the microjets (b) 3D illustration of the cell used for microjet injection (in red)

Fig. 3. Mesh adaptation for microjets modelling. Planar cut of volume mesh is represented in black, nozzle in grey and microjet cell in red.

studies demonstrated that, thanks to the very small time step in the acoustic propagation area (chosen to verify a CFL condition lower than 1 in the flow), the first order implicit time scheme is sufficient for a correct wave propagation and is consequently used in the present study.

#### **2.2 Microjets modelling**

To avoid the necessity of very small sized cells and restricting time step for the microjets representation, a simplified model corresponding to volume sources located in specific cells is employed. The objective of this approach is not to provide an extremely precise and detailed description of the actuators. It is used to investigate the concept of flow injection for noise reduction, which is numerically challenging and even innovative for pulsed actuators. One must furthermore keep in mind that this approach should in the future be used for larger scale, double stream configurations, for which a detailed description of the microjets will not be numerically affordable in the near future due to the complexity of the configurations.

To reproduce the geometric configuration used experimentally, where the microjets are located very close to the nozzle lip, the structured patch has been extended near the jet exit, as illustrated on Fig. 3.

With the present approach, the mass injection and convection due to the microjets is modelled through source terms in the equations of mass (*m*˙ ), momentum (*m*˙ *U* ) and energy (*mE*˙ ) in specific hexahedral cells, identified as the locations of the microjets.

The choice is made here to calibrate the microjets with their velocity and temperature. Velocity vector *U* corresponds to the orientation and velocity of the injected fluid and energy *E* is given by the following formula:

$$E = h\_{ref} + Cp\left(T - T\_{ref}\right) + \mathcal{U}^2/2\tag{2}$$

where *Tref* is the reference temperature, *href* the enthalpy of formation at reference temperature and *Cp* the calorific capacity at constant pressure.

To ensure that the computed velocity and temperature correspond to the desired values, the mass flow rate *m*˙ is determined through an iterative process. Based on the aimed velocity and temperature, an estimation of the initial mass flow rate can be calculated using the fluid velocity, its density and the surface *S* used for the injection: *m*˙ = *ρUS*, where density is approximated by considering that the microjet flow occurs at ambient pressure. The surface can be evaluated from cell geometry and velocity vector: *S* = *U* · *S*/*U* with *S* = ∑*i Si* ·*ni* the sum of downstream exhaust surfaces *Si* of the hexahedral cell used for microjet injection and *ni* their associated outside-oriented normal vectors.

Using this iterative approach, mass flow rate is updated at each iteration until velocity and temperature reach the target values. From the computed surface used for the injection, it is in addition possible to evaluate the equivalent microjet diameter, considering a circular microjet nozzle : *deq* = 2 √*S*/*π*.

#### **2.3 Noise radiation**

6 Will-be-set-by-IN-TECH

Fig. 3. Mesh adaptation for microjets modelling. Planar cut of volume mesh is represented in

studies demonstrated that, thanks to the very small time step in the acoustic propagation area (chosen to verify a CFL condition lower than 1 in the flow), the first order implicit time scheme is sufficient for a correct wave propagation and is consequently used in the present study.

To avoid the necessity of very small sized cells and restricting time step for the microjets representation, a simplified model corresponding to volume sources located in specific cells is employed. The objective of this approach is not to provide an extremely precise and detailed description of the actuators. It is used to investigate the concept of flow injection for noise reduction, which is numerically challenging and even innovative for pulsed actuators. One must furthermore keep in mind that this approach should in the future be used for larger scale, double stream configurations, for which a detailed description of the microjets will not be numerically affordable in the near future due to the complexity of the configurations.

To reproduce the geometric configuration used experimentally, where the microjets are located very close to the nozzle lip, the structured patch has been extended near the jet exit, as

With the present approach, the mass injection and convection due to the microjets is modelled through source terms in the equations of mass (*m*˙ ), momentum (*m*˙ *U* ) and energy (*mE*˙ ) in

The choice is made here to calibrate the microjets with their velocity and temperature. Velocity vector *U* corresponds to the orientation and velocity of the injected fluid and energy *E* is given

where *Tref* is the reference temperature, *href* the enthalpy of formation at reference

*T* − *Tref*

+ *U*2/2 (2)

specific hexahedral cells, identified as the locations of the microjets.

*E* = *href* + *Cp*

temperature and *Cp* the calorific capacity at constant pressure.

(b) 3D illustration of the cell used for

microjet injection (in red)

(a) detail of nozzle exit and mesh

adaptation to model the microjets

black, nozzle in grey and microjet cell in red.

**2.2 Microjets modelling**

illustrated on Fig. 3.

by the following formula:

Instantaneous flow fields provided by the flow simulation and stored on surfaces are used to radiate sound to the microphones. Several integral formulations such as Lighthill (Lighthill, 1952), Kirchhoff (Lyrintzis, 1994) and Ffowcs Williams & Hawkings (FW-H) (Ffowcs Williams & Hawkings, 1969) are available to reconstruct far field noise from volume or surface fluctuations. It is usually preferred to use surface fields to limit the amount of stored data and numerical operations required to perform the noise radiation, which is done in the present study. Lighthill volume integral formulation can nevertheless be fruitful if one wants to identify the noise sources present in the flow (in Lighthill's sense), as done by Perez (Perez et al., 2007).

Based on the extended investigations performed on the topic by Rahier (Rahier et al., 2004), noise radiation is performed in the present case using FW-H porous surface formulation. In their paper, the authors especially demonstrated that this formulation makes it possible to place the control surface rather close to the jet, in a non-uniform flow. The surface position requirement is that the non-uniformities of the flow on the control surface are only due to convected gradients (such as density gradients related to thermal mixing in the case of a hot jet) and are not high sources of noise production (such as instability expanding or vortex pairing). On the contrary, the Kirchhoff method (with density or pressure as input data) is a much less suited tool for acoustic post-processing of jet aerodynamic simulations because of its high sensibility to non acoustic density or pressure gradients (local thermal mixing or vorticity).

Control surfaces are moreover kept open at both extremities. Investigations indeed showed that closing the surface leads in the best case to results similar as with open surface and can provide erroneous results if the control surface is too short, because of the turbulence level on the downstream closing disc. As a consequence of using an open control surface, its axial extent needs to be sufficiently large for a correct radiation at low angles, where the maximum noise is expected. It practically leads to a length of 20 to 25 nozzle diameters.

An axial extent of 25 diameters is used for the present simulations. Surfaces additionally verify 1 ≤ *r*/*D* ≤ 2 at nozzle exit and 3 ≤ *r*/*D* ≤ 4 at the downstream end. Radiation is performed for 4 different surfaces. Collapsing time signals and pressure spectra in the far field ensure that the surfaces enclose all noise sources and provide a limited dissipation for frequencies below a Strouhal number of 0.5.

*xc*/*D u*�

isothermal jet, baseline experiment 7.9 0.135 heated jet, baseline experiment 6.3 0.155

the jet axis and the shear shear.

*rm* = 1.3 × 10−3 for the heated one.

the frequency domain for further analyses.

**3. Aerodynamic results 3.1 Reference simulations**

isothermal jet, baseline 5.2 0.167 0.186 isothermal jet, continuous microjets 5.7 0.159 0.183 isothermal jet, *fF* = 3 kHz pulsed microjets 7.1 0.178 0.211 isothermal jet, *fF* = 9 kHz pulsed microjets 6.6 0.153 0.176 heated jet, baseline 4.1 0.168 0.195 heated jet, continuous microjets 4.7 0.187 0.198

Simulation of Flow Control with Microjets for Subsonic Jet Noise Reduction 87

Table 2. Axial position of the end of the potential core *xc* and peak axial r.m.s. velocity along

The computed injected mass for each continuous microjet is *Qmjet* = 4.8 × 10−4 kg/s. This value is 5% more important than the estimation made at the beginning of the iterative process, based on fluid velocity, estimated density at ambient pressure and injection surface: *Qth* = 4.6 × 10−4 kg/s. The small difference can come from pressure variations at microjets injection. The mass flow rate ratio per microjet *rm* = *Qmjet*/*Qj* is *rm* = 6.7 × 10−4 for the cold jet and

Microphones are located 50 diameters from the nozzle exit. JEAN experimental results, performed at a distance of 30 *D*, are rescaled to 50 *D* using the far field acoustic approximation with 1/*r* pressure amplitude decrease. Integrated pressure levels are computed for frequencies between 300 Hz and 41 kHz. The upper limit is higher than the grid cut-off frequency given in §2.1 because one of the objectives of the approach developed at Onera and used here is to compare simulations to experiments on the experimental frequency bandwidth. The underestimation of the computed OASPL caused by the poorly resolved high frequency levels is nevertheless negligible because the major part of the energy is contained in the low frequency range, below *St* = 0.5, that is accurately resolved in the simulations. The numerical procedure used for all simulations is the following. Every aerodynamic simulation starts from rest and inlet pressure is progressively increased to reach the target value. Computation is then run until the flow is developed in the refined domain, after what surface storage and mean flow averaging begins. The storage is performed during 50 ms of simulated time, which corresponds to 300 *D*/*Uj* convective time units. This duration ensure statistically converged mean flow fields and sufficiently resolved far field pressure spectra in

Evolution of mean axial velocity and turbulent axial velocity along the jet axis are represented on Fig. 4 for the reference simulations. For both temperatures the potential core length, defined as the axial distance from the nozzle exit at which *Ux*/*Uj* = 0.9 and reported on Table 2 is about 30% shorter in the simulation compared to the experiment. Both simulations

moreover overestimate the peak axial r.m.s. velocity on the jet axis.

*x rms*/*Uj* max *u*�

at *r*/*D* = 0 at *r*/*D* = 0.5

*x rms*/*Uj* max

(*x*/*D* > 1)

Table 1. Main jets experimental characteristics.

To end, computed spectra and integrated pressure levels are azimuthally averaged over 48 microphones located around the jet axis. In the present case of an axisymmetric geometry, far field noise is indeed expected to be statistically independent of the azimuth, which is not the case for short-time duration computed signals and can lead to differences of almost 2 dB (Huet et al., 2009).

#### **2.4 Simulated configurations and numerical procedure**

The nozzle used for the simulations is a convergent single stream nozzle with an exhaust diameter *D* = 50 mm. Two main jet configurations, corresponding to isothermal and heated flow, are investigated, which main characteristics are recalled on Table 1. Those configurations correspond to the baseline and are used as reference when investigating the effect of microjets on flow and noise.

Based on grid construction and flow characteristics, the aerodynamic time step is set to Δ*taero* = 10−6 s, which corresponds to a CFL criterion lower than 1 in most of the computational domain. Surface storage is performed every 10 iterations of the flow simulation, Δ*tacou* = 10−5 s. It corresponds to a maximum frequency of 50 kHz, which ensures that every frequency correctly resolved by the LES will be properly radiated using the integral formulation.

The control of the jet is performed with the use of 12 microjets regularly positioned around the nozzle lip, just after the exhaust area, with an impinging angle of 45 degrees relative to the jet axis. Continuous microjets are computed for both main jets. Pulsed microjets, for which simulations are run as exploratory computations only, are just performed for the isothermal configuration with all actuators in phase, which corresponds to the axisymmetric mode *m* = 0. Their flow and geometrical characteristics are identical to those used for the continuous microjets; time modulation of the source terms is a periodized crenel, whose value is set to 1 during 1/3rd of the period and 0 otherwise. This modulation is a first representation of the plasma synthetic jets developed at Onera for flow and noise control (Caruana et al., 2009; Hardy et al., 2010). The two forcing frequencies *fF* = 3 kHz and 9 kHz are considered for the simulations. They are chosen following the experimental results of Samimy (Samimy et al., 2007a) using localized arc filament plasma actuators and who observed a broadband noise increase for the low frequency excitation and a reduction for the high one.

For all simulations, microjets velocity and temperature are set to 300 m/s and 288 K respectively. Those values correspond to the ones used for the main isothermal jet.

Equivalent microjet diameter is *deq* = 1.27 mm. This diameter is comparable to the 1 mm diameter of the actuators used at Institut PPRIME during the acoustic test campaign of the project and by Castelain (Castelain et al., 2007; 2008) for aerodynamic and acoustic measurements performed in a separate framework. The distance between the microjets and the main jet shear layer is 0.40 *deq*. It ensures that the microjets flow spreading is very limited before reaching the jet flow and that its action on the shear layer is not negligible.

8 Will-be-set-by-IN-TECH

*Ts* (K) *Uj* (m/s) *Ma Mj ReD Qj* (kg/s) 288 306.65 0.90 0.90 1.0 × 106 0.723 576 306.65 0.90 0.64 3.2 × 105 0.362

To end, computed spectra and integrated pressure levels are azimuthally averaged over 48 microphones located around the jet axis. In the present case of an axisymmetric geometry, far field noise is indeed expected to be statistically independent of the azimuth, which is not the case for short-time duration computed signals and can lead to differences of almost 2 dB

The nozzle used for the simulations is a convergent single stream nozzle with an exhaust diameter *D* = 50 mm. Two main jet configurations, corresponding to isothermal and heated flow, are investigated, which main characteristics are recalled on Table 1. Those configurations correspond to the baseline and are used as reference when investigating the effect of microjets

Based on grid construction and flow characteristics, the aerodynamic time step is set to Δ*taero* = 10−6 s, which corresponds to a CFL criterion lower than 1 in most of the computational domain. Surface storage is performed every 10 iterations of the flow simulation, Δ*tacou* = 10−5 s. It corresponds to a maximum frequency of 50 kHz, which ensures that every frequency correctly resolved by the LES will be properly radiated using the integral

The control of the jet is performed with the use of 12 microjets regularly positioned around the nozzle lip, just after the exhaust area, with an impinging angle of 45 degrees relative to the jet axis. Continuous microjets are computed for both main jets. Pulsed microjets, for which simulations are run as exploratory computations only, are just performed for the isothermal configuration with all actuators in phase, which corresponds to the axisymmetric mode *m* = 0. Their flow and geometrical characteristics are identical to those used for the continuous microjets; time modulation of the source terms is a periodized crenel, whose value is set to 1 during 1/3rd of the period and 0 otherwise. This modulation is a first representation of the plasma synthetic jets developed at Onera for flow and noise control (Caruana et al., 2009; Hardy et al., 2010). The two forcing frequencies *fF* = 3 kHz and 9 kHz are considered for the simulations. They are chosen following the experimental results of Samimy (Samimy et al., 2007a) using localized arc filament plasma actuators and who observed a broadband noise

For all simulations, microjets velocity and temperature are set to 300 m/s and 288 K

Equivalent microjet diameter is *deq* = 1.27 mm. This diameter is comparable to the 1 mm diameter of the actuators used at Institut PPRIME during the acoustic test campaign of the project and by Castelain (Castelain et al., 2007; 2008) for aerodynamic and acoustic measurements performed in a separate framework. The distance between the microjets and the main jet shear layer is 0.40 *deq*. It ensures that the microjets flow spreading is very limited

increase for the low frequency excitation and a reduction for the high one.

respectively. Those values correspond to the ones used for the main isothermal jet.

before reaching the jet flow and that its action on the shear layer is not negligible.

Table 1. Main jets experimental characteristics.

**2.4 Simulated configurations and numerical procedure**

(Huet et al., 2009).

on flow and noise.

formulation.

Table 2. Axial position of the end of the potential core *xc* and peak axial r.m.s. velocity along the jet axis and the shear shear.

The computed injected mass for each continuous microjet is *Qmjet* = 4.8 × 10−4 kg/s. This value is 5% more important than the estimation made at the beginning of the iterative process, based on fluid velocity, estimated density at ambient pressure and injection surface: *Qth* = 4.6 × 10−4 kg/s. The small difference can come from pressure variations at microjets injection. The mass flow rate ratio per microjet *rm* = *Qmjet*/*Qj* is *rm* = 6.7 × 10−4 for the cold jet and *rm* = 1.3 × 10−3 for the heated one.

Microphones are located 50 diameters from the nozzle exit. JEAN experimental results, performed at a distance of 30 *D*, are rescaled to 50 *D* using the far field acoustic approximation with 1/*r* pressure amplitude decrease. Integrated pressure levels are computed for frequencies between 300 Hz and 41 kHz. The upper limit is higher than the grid cut-off frequency given in §2.1 because one of the objectives of the approach developed at Onera and used here is to compare simulations to experiments on the experimental frequency bandwidth. The underestimation of the computed OASPL caused by the poorly resolved high frequency levels is nevertheless negligible because the major part of the energy is contained in the low frequency range, below *St* = 0.5, that is accurately resolved in the simulations.

The numerical procedure used for all simulations is the following. Every aerodynamic simulation starts from rest and inlet pressure is progressively increased to reach the target value. Computation is then run until the flow is developed in the refined domain, after what surface storage and mean flow averaging begins. The storage is performed during 50 ms of simulated time, which corresponds to 300 *D*/*Uj* convective time units. This duration ensure statistically converged mean flow fields and sufficiently resolved far field pressure spectra in the frequency domain for further analyses.

#### **3. Aerodynamic results**

#### **3.1 Reference simulations**

Evolution of mean axial velocity and turbulent axial velocity along the jet axis are represented on Fig. 4 for the reference simulations. For both temperatures the potential core length, defined as the axial distance from the nozzle exit at which *Ux*/*Uj* = 0.9 and reported on Table 2 is about 30% shorter in the simulation compared to the experiment. Both simulations moreover overestimate the peak axial r.m.s. velocity on the jet axis.

Fig. 6. Illustration of the vortex pairing phenomenon (arrow) observed for the isothermal

With the considered Reynolds numbers, the shear layer is expected to be fully turbulent for the isothermal jet or at least nominally turbulent for the heated jet, using the terminology proposed by Zaman (Zaman, 1985). Its initially laminar state might be explained by the unperturbed imposed inflow as well as the lack of resolution of the boundary layers of the nozzle. This low discretization would not allow sufficient resolution of the perturbations that are expected to develop and to trigger the turbulent shear layer development. Similar observations have been made by Vuillot (Vuillot et al., 2011) on a double stream nozzle with

Simulation of Flow Control with Microjets for Subsonic Jet Noise Reduction 89

An illustration of the initially laminar shear layer development is visible on Fig. 6 for the isothermal jet. One can first observe on the instantaneous density field represented at three different instants a vortex shedding phenomenon, corresponding to the green dots along the shear layer. This shedding is followed by multiple vortex pairings occurring 1 diameter or more after the nozzle exit, one of them being visible in the pictures at the end of the arrow. These phenomena are characteristic of laminar to turbulent transition in the mixing layer,

In addition to the initially laminar state of the computed jets, the overestimation of the turbulence observed on Fig. 4 (b) and on Fig. 5 might also come partly from a too fast coarsening of the grid in the axial direction. Indeed, a coarse grid tends to overestimate the large turbulent structures and thus the turbulent kinetic energy, leading to a too fast growth

Despite those discrepancies between the simulations and the experiments, the effect of temperature on mean and r.m.s. velocities is well reproduced by the simulations, with a reduction of the potential core length of 20% with increasing temperature for both measurements and computations. The consequence of this core length shortening is the reduction of the axial distance at which the peak axial r.m.s. velocity occurs. The peak axial r.m.s. level increase observed experimentally on the jet axis with increasing temperature is nevertheless not clearly visible in the simulations, with only a 0.5% raise compared to 20% in the experiments. Higher turbulence levels for the heated jet are however observed

No aerodynamic measurements being available for the simulated configurations with microjets, investigations presented hereafter are performed comparing baseline and

explaining the peak axial r.m.s. velocity observed on Fig. 5 for the computations.

of the shear layers and a shortening of the potential core.

controlled simulation with available literature results.

numerically along the shear layer.

**3.2 Continuous microjets**

simulated jet. Coloured map represents density field at 3 different instants.

external flow.

Fig. 4. Experimental and simulated axial (a) mean and (b) r.m.s. velocity on jet axis for isothermal (blue) and heated (red) baseline configurations. � experiments; – simulations.

Fig. 5. Evolution of the axial r.m.s. velocity *u*� *rms* along the nozzle lip. Simulations: – isothermal and – heated baseline configurations. Measurements: ♦ Husain and Hussain for an initially turbulent shear layer (Husain & Hussain, 1979) and ∇ Fleury et al. for a Mach 0.9 jet at *ReD* = 7.7 × 105 (Fleury et al., 2008).

This numerical underestimation of the potential core length has already been observed in previous computations of the same geometry (Andersson et al., 2005; Bodard et al., 2009; Bogey & Bailly, 2006) and might come partly, in the present computations, from different initial conditions between experiments and simulations. Fig. 4 (b) illustrates the differences between experimental and simulated turbulence levels at nozzle exit, experiments exhibiting the presence of velocity perturbations which are not present in the computations. These very low computed turbulence levels are a consequence of the unperturbed inflow imposed on the nozzle inlet boundary and may be the reason of the discrepancies observed. Bogey (Bogey & Bailly, 2010) indeed numerically demonstrated for a *Mj* = 0.9 and *ReD* = 105 jet that the presence of disturbances in the nozzle increase the potential core length and lower the turbulence levels on the jet axis.

Simulated turbulence profiles along the nozzle lip reproduced on Fig. 5 exhibit a peak value for *x*/*D* ∼ 4 and *x*/*D* ∼ 3 respectively for the isothermal and the heated jet, whereas those turbulence levels increase nearly monotonically for experimental fully turbulent jets also reported on the figure. These peak r.m.s. values might indicate a strong transition of the initially laminar jet mixing layer in the computations.

10 Will-be-set-by-IN-TECH

(a) axial mean velocity (b) axial r.m.s. velocity

isothermal and – heated baseline configurations. Measurements: ♦ Husain and Hussain for an initially turbulent shear layer (Husain & Hussain, 1979) and ∇ Fleury et al. for a Mach 0.9

This numerical underestimation of the potential core length has already been observed in previous computations of the same geometry (Andersson et al., 2005; Bodard et al., 2009; Bogey & Bailly, 2006) and might come partly, in the present computations, from different initial conditions between experiments and simulations. Fig. 4 (b) illustrates the differences between experimental and simulated turbulence levels at nozzle exit, experiments exhibiting the presence of velocity perturbations which are not present in the computations. These very low computed turbulence levels are a consequence of the unperturbed inflow imposed on the nozzle inlet boundary and may be the reason of the discrepancies observed. Bogey (Bogey & Bailly, 2010) indeed numerically demonstrated for a *Mj* = 0.9 and *ReD* = 105 jet that the presence of disturbances in the nozzle increase the potential core length and lower the

Simulated turbulence profiles along the nozzle lip reproduced on Fig. 5 exhibit a peak value for *x*/*D* ∼ 4 and *x*/*D* ∼ 3 respectively for the isothermal and the heated jet, whereas those turbulence levels increase nearly monotonically for experimental fully turbulent jets also reported on the figure. These peak r.m.s. values might indicate a strong transition of the

*rms* along the nozzle lip. Simulations: –

Fig. 4. Experimental and simulated axial (a) mean and (b) r.m.s. velocity on jet axis for isothermal (blue) and heated (red) baseline configurations. � experiments; – simulations.

Fig. 5. Evolution of the axial r.m.s. velocity *u*�

jet at *ReD* = 7.7 × 105 (Fleury et al., 2008).

turbulence levels on the jet axis.

initially laminar jet mixing layer in the computations.

Fig. 6. Illustration of the vortex pairing phenomenon (arrow) observed for the isothermal simulated jet. Coloured map represents density field at 3 different instants.

With the considered Reynolds numbers, the shear layer is expected to be fully turbulent for the isothermal jet or at least nominally turbulent for the heated jet, using the terminology proposed by Zaman (Zaman, 1985). Its initially laminar state might be explained by the unperturbed imposed inflow as well as the lack of resolution of the boundary layers of the nozzle. This low discretization would not allow sufficient resolution of the perturbations that are expected to develop and to trigger the turbulent shear layer development. Similar observations have been made by Vuillot (Vuillot et al., 2011) on a double stream nozzle with external flow.

An illustration of the initially laminar shear layer development is visible on Fig. 6 for the isothermal jet. One can first observe on the instantaneous density field represented at three different instants a vortex shedding phenomenon, corresponding to the green dots along the shear layer. This shedding is followed by multiple vortex pairings occurring 1 diameter or more after the nozzle exit, one of them being visible in the pictures at the end of the arrow. These phenomena are characteristic of laminar to turbulent transition in the mixing layer, explaining the peak axial r.m.s. velocity observed on Fig. 5 for the computations.

In addition to the initially laminar state of the computed jets, the overestimation of the turbulence observed on Fig. 4 (b) and on Fig. 5 might also come partly from a too fast coarsening of the grid in the axial direction. Indeed, a coarse grid tends to overestimate the large turbulent structures and thus the turbulent kinetic energy, leading to a too fast growth of the shear layers and a shortening of the potential core.

Despite those discrepancies between the simulations and the experiments, the effect of temperature on mean and r.m.s. velocities is well reproduced by the simulations, with a reduction of the potential core length of 20% with increasing temperature for both measurements and computations. The consequence of this core length shortening is the reduction of the axial distance at which the peak axial r.m.s. velocity occurs. The peak axial r.m.s. level increase observed experimentally on the jet axis with increasing temperature is nevertheless not clearly visible in the simulations, with only a 0.5% raise compared to 20% in the experiments. Higher turbulence levels for the heated jet are however observed numerically along the shear layer.

#### **3.2 Continuous microjets**

No aerodynamic measurements being available for the simulated configurations with microjets, investigations presented hereafter are performed comparing baseline and controlled simulation with available literature results.

(a) isothermal jet (b) heated jet

Simulation of Flow Control with Microjets for Subsonic Jet Noise Reduction 91

Fig. 9. Axial r.m.s. velocity distribution for (top) baseline and (bottom) continuous microjets

(a) isothermal jet (b) heated jet

(a) isothermal jet (b) heated jet

Fig. 11. Turbulent kinetic energy distribution for (top) baseline and (bottom) continuous

Fig. 10. Radial r.m.s. velocity distribution for (top) baseline and (bottom) continuous

microjets configurations. Plane located through microjet position.

microjets configurations. Plane located through microjet position.

configurations. Plane located through microjet position.

Fig. 7. Mean axial velocity computed at *x*/*D* = 1 for the baseline (left) and continuous microjets (right) isothermal configurations. Arrows represent the position of the microjets.

Fig. 8. Mean axial velocity distribution for (top) baseline and (bottom) continuous microjets configurations. Plane located through microjet position.

An illustration of the efficiency of the microjets for modifying main jet mean flow is visible on Fig. 7. It is clearly visible on this figure that the actuators corrugate the jet azimuthally by locally decreasing the axial velocity downstream of the microjets. These corrugations are limited in space to the nozzle vicinity and vanish more downstream after 3 diameters, as experimentally observed (Alkislar et al., 2007; Arakeri et al., 2003; Castelain et al., 2007). They are a consequence of the difference in axial velocity between the microjets and the main jet that is expected to generate flow mixing.

Cartographies of mean axial velocity, reproduced on Fig. 8, illustrate a stretching of the jet with the presence of the microjets. This stretching is accompanied by an increase of the potential core length of 10% and 15%, respectively for the isothermal and heated configurations, especially visible on Fig. 12 (a). This result is consistent with the experimental observations of Castelain et al. and Arakeri et al., the latter observing an increase of 25% caused by the control.

12 Will-be-set-by-IN-TECH

Fig. 7. Mean axial velocity computed at *x*/*D* = 1 for the baseline (left) and continuous microjets (right) isothermal configurations. Arrows represent the position of the microjets.

(a) isothermal jet (b) heated jet

Fig. 8. Mean axial velocity distribution for (top) baseline and (bottom) continuous microjets

An illustration of the efficiency of the microjets for modifying main jet mean flow is visible on Fig. 7. It is clearly visible on this figure that the actuators corrugate the jet azimuthally by locally decreasing the axial velocity downstream of the microjets. These corrugations are limited in space to the nozzle vicinity and vanish more downstream after 3 diameters, as experimentally observed (Alkislar et al., 2007; Arakeri et al., 2003; Castelain et al., 2007). They are a consequence of the difference in axial velocity between the microjets and the main jet

Cartographies of mean axial velocity, reproduced on Fig. 8, illustrate a stretching of the jet with the presence of the microjets. This stretching is accompanied by an increase of the potential core length of 10% and 15%, respectively for the isothermal and heated configurations, especially visible on Fig. 12 (a). This result is consistent with the experimental observations of Castelain et al. and Arakeri et al., the latter observing an increase of 25%

configurations. Plane located through microjet position.

that is expected to generate flow mixing.

caused by the control.

Fig. 9. Axial r.m.s. velocity distribution for (top) baseline and (bottom) continuous microjets configurations. Plane located through microjet position.

Fig. 10. Radial r.m.s. velocity distribution for (top) baseline and (bottom) continuous microjets configurations. Plane located through microjet position.

Fig. 11. Turbulent kinetic energy distribution for (top) baseline and (bottom) continuous microjets configurations. Plane located through microjet position.

(a) *fF* = 3 kHz pulsed microjets (b) *fF* = 9 kHz pulsed microjets

Simulation of Flow Control with Microjets for Subsonic Jet Noise Reduction 93

Fig. 14. Turbulent kinetic energy distribution for (top) baseline and (bottom) pulsed microjets

downstream from the nozzle exit compared to the baseline configurations, as visible on Fig. 11. On this figure is also observed a high turbulence level just downstream of the microjets injection, that comes essentially from the contribution of the axial turbulent velocity. This increase is not observed in the experiments and can be attributed in the present simulations

Despite the initial state of the shear layer that is not fully turbulent in the simulations, which leads especially to some differences on flow characteristics with the experiments, all the aerodynamic simulated results with continuous microjets presented in the section above compare rather well with the experimental data. This tends to illustrate that the actions of the actuators on the main jet are correctly reproduced in the computations. A further investigation on the efficiency of such a type of control is thus performed by considering pulsed microjets. It has to be recalled nevertheless that the modelled pulsed actuators do not intend to precisely reproduce the existing plasma synthetic jets developed at Onera (Caruana et al., 2009; Hardy et al., 2010), but are used to illustrate the capacity of the numerical code to reproduce the

Fig. 12 illustrates the more important potential core length increase obtained with the pulsed control, compared to the continuous one. This core increase is related for both simulations to the reduction of the axial r.m.s. velocity along the shear layer, the more important reduction being observed for the high frequency excitation. This turbulence reduction observed for the *fF* = 9 kHz forcing is qualitatively similar to the one occurring with continuous microjets, for both axial and radial turbulent velocities, and is simply more pronounced, as visible on Fig. 13. This is clearly visible when looking at the turbulent kinetic energy distribution,

A different evolution of the turbulence is observed for the low frequency forced jet. Velocity fluctuations present in this case a very large increase a few diameters after the nozzle exit, the peak turbulent kinetic energy being localized about 1.5 *D* downstream of the jet exit, see Fig. 14 (a). The oscillations observed on this figure only come from the too short time duration in the averaging and are caused by the strong response of the jet to the excitation, visible on Fig.

15 (c), and which was not observed with continuous or *fF* = 9 kHz pulsed control.

configurations. Plane located through microjet position.

to the initially laminar shear layer.

**3.3 Pulsed microjets**

action of such microjets.

reproduced on Fig. 14 (b).

Fig. 12. Simulated axial (a) mean and (b) r.m.s. velocity on jet axis. Isothermal jets: – baseline; – – continuous microjets; – *fF* = 3 kHz pulsed microjets; – *fF* = 9 kHz pulsed microjets. Heated jets: – baseline; – – continuous microjets.

Fig. 13. Evolution of the r.m.s. velocities along the nozzle lip. The legend is the same as on Fig. 12.

As experimentally expected, the presence of the microjets also leads to a reduction of the peak axial r.m.s. velocity on the jet axis for the isothermal jet, see Fig. 12 (b). This peak value is however increased for the heated jet and is a consequence of a spot of high axial velocity fluctuations appearing near the jet axis for *x*/*D* = 8, as visible on Fig. 9. Except for this spot, no significant modification of the peak value is observed for the heated jet, its axial location being shifted downstream compared to the baseline configuration because of the more important jet stretching relative to the isothermal simulations.

Along the shear layer and for the isothermal jet, actuators lead to a reduction of the turbulence in the first diameters after the nozzle exit, see Fig. 13. This result is similar with the observations of Castelain (Castelain, 2006). It nevertheless slightly differs from the results of Arakeri (Arakeri et al., 2003) who measured a reduction of the peak axial r.m.s. velocity that is not reproduced by the simulations. Similar flow modifications are observed for the heated jet, except for the peak radial r.m.s. velocity that is now reduced in the computations.

Such modifications in the turbulent velocities numerically lead, for both temperatures, to a reduction of the peak turbulent kinetic energy in the jet plume, that appears further

Fig. 14. Turbulent kinetic energy distribution for (top) baseline and (bottom) pulsed microjets configurations. Plane located through microjet position.

downstream from the nozzle exit compared to the baseline configurations, as visible on Fig. 11. On this figure is also observed a high turbulence level just downstream of the microjets injection, that comes essentially from the contribution of the axial turbulent velocity. This increase is not observed in the experiments and can be attributed in the present simulations to the initially laminar shear layer.

#### **3.3 Pulsed microjets**

14 Will-be-set-by-IN-TECH

(a) axial mean velocity (b) axial r.m.s. velocity

Fig. 12. Simulated axial (a) mean and (b) r.m.s. velocity on jet axis. Isothermal jets: – baseline; – – continuous microjets; – *fF* = 3 kHz pulsed microjets; – *fF* = 9 kHz pulsed microjets.

(a) axial r.m.s. velocity (b) radial r.m.s. velocity

As experimentally expected, the presence of the microjets also leads to a reduction of the peak axial r.m.s. velocity on the jet axis for the isothermal jet, see Fig. 12 (b). This peak value is however increased for the heated jet and is a consequence of a spot of high axial velocity fluctuations appearing near the jet axis for *x*/*D* = 8, as visible on Fig. 9. Except for this spot, no significant modification of the peak value is observed for the heated jet, its axial location being shifted downstream compared to the baseline configuration because of

Along the shear layer and for the isothermal jet, actuators lead to a reduction of the turbulence in the first diameters after the nozzle exit, see Fig. 13. This result is similar with the observations of Castelain (Castelain, 2006). It nevertheless slightly differs from the results of Arakeri (Arakeri et al., 2003) who measured a reduction of the peak axial r.m.s. velocity that is not reproduced by the simulations. Similar flow modifications are observed for the heated jet, except for the peak radial r.m.s. velocity that is now reduced in the computations. Such modifications in the turbulent velocities numerically lead, for both temperatures, to a reduction of the peak turbulent kinetic energy in the jet plume, that appears further

the more important jet stretching relative to the isothermal simulations.

Fig. 13. Evolution of the r.m.s. velocities along the nozzle lip. The legend is the same as on

Heated jets: – baseline; – – continuous microjets.

Fig. 12.

Despite the initial state of the shear layer that is not fully turbulent in the simulations, which leads especially to some differences on flow characteristics with the experiments, all the aerodynamic simulated results with continuous microjets presented in the section above compare rather well with the experimental data. This tends to illustrate that the actions of the actuators on the main jet are correctly reproduced in the computations. A further investigation on the efficiency of such a type of control is thus performed by considering pulsed microjets. It has to be recalled nevertheless that the modelled pulsed actuators do not intend to precisely reproduce the existing plasma synthetic jets developed at Onera (Caruana et al., 2009; Hardy et al., 2010), but are used to illustrate the capacity of the numerical code to reproduce the action of such microjets.

Fig. 12 illustrates the more important potential core length increase obtained with the pulsed control, compared to the continuous one. This core increase is related for both simulations to the reduction of the axial r.m.s. velocity along the shear layer, the more important reduction being observed for the high frequency excitation. This turbulence reduction observed for the *fF* = 9 kHz forcing is qualitatively similar to the one occurring with continuous microjets, for both axial and radial turbulent velocities, and is simply more pronounced, as visible on Fig. 13. This is clearly visible when looking at the turbulent kinetic energy distribution, reproduced on Fig. 14 (b).

A different evolution of the turbulence is observed for the low frequency forced jet. Velocity fluctuations present in this case a very large increase a few diameters after the nozzle exit, the peak turbulent kinetic energy being localized about 1.5 *D* downstream of the jet exit, see Fig. 14 (a). The oscillations observed on this figure only come from the too short time duration in the averaging and are caused by the strong response of the jet to the excitation, visible on Fig. 15 (c), and which was not observed with continuous or *fF* = 9 kHz pulsed control.

In the present simulation, the hyperbolic-tangent velocity profiles well fit the simulated profiles in the first jet diameters. More downstream some differences can be observed between simulated and analytical profiles, that might lead to an underestimation of the calculated frequencies. The modelling however gives a correct order of those two characteristic

Simulation of Flow Control with Microjets for Subsonic Jet Noise Reduction 95

The evolution of the momentum thickness in the first diameters after the nozzle exit is represented on Fig. 16. On this figure are also visible the most unstable shear layer frequency and the neutral disturbance frequency. It is clearly observed that both frequencies of 3 kHz and 9 kHz correspond to most unstable frequencies within the first diameter after the nozzle exit. However, the neutral frequency is observed just downstream for the 9 kHz frequency, which can explain that no destabilization is noticed: each instability appearing in the jet is nearly instantaneously damped. This damping is theoretically expected further in the jet for 3 kHz perturbations, which gives the time for the shear layer instabilities to develop. This result is coherent with the illustration of the instantaneous axial velocity on Fig. 15 (c), where strong

Noise modifications caused by temperature increase and microjets are presented in this section. Simulations with continuous control are compared to measurements provided in the frame of the study by French institute PPRIME for angles above 50 degrees. Experimental data from European project JEAN, also performed by institute PPRIME, are nevertheless preferred for temperature effect investigation, as they provide noise levels below 50 degrees. Comparisons have been made to ensure that both experimental campaigns provide similar

The effect of temperature on jet noise has been widely investigated experimentally (Tanna, 1977; Tanna et al., 1975; Viswanathan, 2004). It has been observed that, for a fixed acoustic Mach number *Ma* = *Uj*/*c*0, where *c*0 is the sound velocity at ambient temperature, the radiated noise directly depends on the jet temperature. For *Ma* < 0.7, increasing the jet temperature increases the noise; no effect is observed for *Ma* = 0.7 and the noise is reduced with increasing temperature for *Ma* > 0.7. For this last case corresponding to the configurations presented here, the spectral modifications are a decrease of the medium and high frequency levels when temperature rises, while no modifications are visible for the low

Experimental data reproduced on Fig. 17 exhibit spectral modifications with temperature increase that conform to literature observations. One can nevertheless notice a slight noise increase around 1 kHz at 60 degrees, which was not mentioned in the above-cited published

Simulated power spectral densities are illustrated on Fig. 18. Except for very low frequencies, where jet heating reduces the noise levels, calculations reproduce rather well the effect of temperature experimentally described. The decrease of medium and high frequency levels with increasing temperature is especially well observed, as well as the small level increase for

oscillations are observed with the *fF* = 3 kHz pulsed control for *x*/*D* = 1.

frequencies of the shear layer.

**4. Acoustic results**

**4.1 Temperature effect**

frequency part of the spectra.

the heated jet near 1 kHz at 60 degrees.

results.

results.

Fig. 15. Instantaneous axial velocity distribution on *y* = 0 plane. Isothermal jet simulations.

Fig. 16. Evolution of the (–) momentum thickness, (– –) most unstable shear layer frequency and (– ·) neutral disturbance frequency. Baseline isothermal jet simulation.

To explain this distinct response of the flow to the *fF* = 3 kHz pulsed excitation, a jet stability analysis is performed on the baseline configuration, using Michalke's theoretical work on the stability of inviscid shear layers (Michalke, 1965). In this theory, the radial profile of the axial velocity is analytically represented with an hyperbolic-tangent profile:

$$\mathcal{U}I(r) = \frac{\mathcal{U}\_j}{2} \left( 1 + \tanh\left(\frac{r}{2\delta\_\theta}\right) \right) \tag{3}$$

where *r* represents the radial distance from the axis and *δθ* the momentum thickness. The most unstable shear layer frequency is then given by *f* = 0.0165 · *Uj*/*δθ* and the neutral disturbance frequency by *f* = 0.0400 · *Uj*/*δθ* . This last frequency corresponds to the threshold value for which the excited shear layer modes are stable and thus do not lead to a destabilization of the jet.

In the present simulation, the hyperbolic-tangent velocity profiles well fit the simulated profiles in the first jet diameters. More downstream some differences can be observed between simulated and analytical profiles, that might lead to an underestimation of the calculated frequencies. The modelling however gives a correct order of those two characteristic frequencies of the shear layer.

The evolution of the momentum thickness in the first diameters after the nozzle exit is represented on Fig. 16. On this figure are also visible the most unstable shear layer frequency and the neutral disturbance frequency. It is clearly observed that both frequencies of 3 kHz and 9 kHz correspond to most unstable frequencies within the first diameter after the nozzle exit. However, the neutral frequency is observed just downstream for the 9 kHz frequency, which can explain that no destabilization is noticed: each instability appearing in the jet is nearly instantaneously damped. This damping is theoretically expected further in the jet for 3 kHz perturbations, which gives the time for the shear layer instabilities to develop. This result is coherent with the illustration of the instantaneous axial velocity on Fig. 15 (c), where strong oscillations are observed with the *fF* = 3 kHz pulsed control for *x*/*D* = 1.

#### **4. Acoustic results**

16 Will-be-set-by-IN-TECH

(a) baseline (b) continuous microjets

(c) *fF* = 3 kHz pulsed microjets (d) *fF* = 9 kHz pulsed microjets

Fig. 15. Instantaneous axial velocity distribution on *y* = 0 plane. Isothermal jet simulations.

Fig. 16. Evolution of the (–) momentum thickness, (– –) most unstable shear layer frequency

To explain this distinct response of the flow to the *fF* = 3 kHz pulsed excitation, a jet stability analysis is performed on the baseline configuration, using Michalke's theoretical work on the stability of inviscid shear layers (Michalke, 1965). In this theory, the radial profile of the axial

1 + *tanh*

where *r* represents the radial distance from the axis and *δθ* the momentum thickness. The most unstable shear layer frequency is then given by *f* = 0.0165 · *Uj*/*δθ* and the neutral disturbance frequency by *f* = 0.0400 · *Uj*/*δθ* . This last frequency corresponds to the threshold value for which the excited shear layer modes are stable and thus do not lead to a destabilization of the

 *r* 2*δθ* (3)

and (– ·) neutral disturbance frequency. Baseline isothermal jet simulation.

velocity is analytically represented with an hyperbolic-tangent profile:

*U*(*r*) =

jet.

*Uj* 2  Noise modifications caused by temperature increase and microjets are presented in this section. Simulations with continuous control are compared to measurements provided in the frame of the study by French institute PPRIME for angles above 50 degrees. Experimental data from European project JEAN, also performed by institute PPRIME, are nevertheless preferred for temperature effect investigation, as they provide noise levels below 50 degrees. Comparisons have been made to ensure that both experimental campaigns provide similar results.

#### **4.1 Temperature effect**

The effect of temperature on jet noise has been widely investigated experimentally (Tanna, 1977; Tanna et al., 1975; Viswanathan, 2004). It has been observed that, for a fixed acoustic Mach number *Ma* = *Uj*/*c*0, where *c*0 is the sound velocity at ambient temperature, the radiated noise directly depends on the jet temperature. For *Ma* < 0.7, increasing the jet temperature increases the noise; no effect is observed for *Ma* = 0.7 and the noise is reduced with increasing temperature for *Ma* > 0.7. For this last case corresponding to the configurations presented here, the spectral modifications are a decrease of the medium and high frequency levels when temperature rises, while no modifications are visible for the low frequency part of the spectra.