Numerical simulations of a subsonic round jet with transverse acoustic and mechanical forcing

. The results of numerical experiments on the bifurcation effects for a free round jet caused by acoustic or vibrational excitations are presented. The three-dimensional distributions of the instantaneous and time-averaged velocity vector components have been obtained and analyzed. The effects of Strouhal and Reynolds numbers on the jet splitting angle, as well as those of the amplitude and type of jet perturbations is discussed in such a comparative parametric study.


Introduction
The interest in methods to control jet flows at different conditions is growing due to the demand in technical applications.The practical needs to reduce jet noise, fuel consumption, and emission of harmful impurities from the operation of industrial devices are of key relevance and require developing an integrated approach to the solutions of these issues related to jet control.
At relatively low jet excitation frequency, mixing is enhanced [1,2], leading to heat transfer intensification, which is applicable, for example, to cool heated surfaces of small-sized devices in electronics.The acoustic or mechanical forcing on the flow region near the jet nozzle enhances the jet expansion, causes its meandering and splitting.In [1][2][3][4][5][6], the main results of physical and numerical experiments on the jet branching during its excitation performed by different ways are mentioned.
The present study shows first the computation results for a free round jet excited by the transverse acoustic field from a single-mode sound source placed at the lateral boundary as in the laboratory experiments [3][4], in comparison with those by the transverse mechanical vibration of the inlet wall section as in the physical experiments [7] and numerical studies [5,[8][9][10].It is of interest here to check the limits of physical parameters, e.g.Strouhal number and forcing amplitude, for the jet splitting effects at relatively low Reynolds numbers, relating to the measurement conditions in [3][4].

Computation setup
The numerical solution of the Navier -Stokes equations (by DNS where no turbulence models are introduced) for a flow including a free jet and a sound source is performed using the finite volume method by means of the Ansys Fluent package.A round air jet, outflowing at uniform velocity U into the open air space from a circular hole (centered at x = y = z = 0) of diameter D at Reynolds number Re (= UD/ν) = 500, is simulated numerically (Fig. 1, a).The computational domain sizes (50D × 20D × 20D) and the number of non-uniform grid cells in the x-y-z directions have been chosen taking into account the problem setup in previous numerical studies [5,[9][10][11][12][13][14].For the transverse acoustic forcing, the sound source centered at x/D = 2.5, y/D = 0, z/D = 10 is localized in the square area 5D × 5D at the lateral boundary A (Fig. 1, b).At t ≥ 0 it generates the pressure oscillation p' = a × sin (2π•f•t) at frequency f, with varied Strouhal number 0.01 ≤ St (= fD/U) ≤ 0.5, and at varied amplitude a ≤ 25 Pa.At t < 0 in the domain we have a developed flow without forcing, i.e. a = 0.At t ≥ 0 such a single jet is gradually transformed to a bifurcating jet (Fig. 2) at specific combinations of forcing parameters.To simulate a round jet with vibrational excitation, the Navier -Stokes equations are solved by means of the OpenFOAM code implementing the reactingFoam solver at moderate Reynolds numbers (50 ≤ Re ≤ 750) [5,9,10] by DNS too where turbulence models are not used.Again, a submerged jet comes into a rectangular domain through a round hole in the inlet wall (Fig. 1, a).The computation grid has the uniform fine-mesh area near the jet inlet, then coarsens in the radial and axial directions.The computation domain sizes and grid resolutions are chosen based on a series of preliminary runs [5,9,10] and are similar to that of [11][12][13][14].The uniform velocity, u(x = 0) = U at r < R and u(x = 0) = 0 at r ≥ R (where R = D/2 is the hole radius), is assigned at inlet as in [5].The harmonic transverse motion of the inlet boundary x = 0, which action on a jet flow is similar to that of acoustic forcing [1][2][3][4], is introduced as follows: where z* includes the transverse displacement Z(t) of the inlet wall, f and Z0 are the excitation frequency (Hz) and its amplitude.Z0 = 0 corresponds to the motionless inlet.

Results
First, the jet behavior is studied for Re = 500 under the acoustic forcing at 'optimal' Strouhal number St = 0.07 with pressure amplitude a = 15 Pa corresponding to the transverse velocity amplitude V0/U = 0.010 at the jet axis point (x = 2.5D, y = z = 0) located normally to the sound source center.The jet splitting development stages are seen as follows from the instantaneous plots for the horizontal velocity vector component u/U (see Fig. 2, where the u scale is the same as in Fig. 3): the jet begins to meander at distance x > 5D and expands (Fig. 2, b), then splits into two branches (Fig. 2, c-f).At large times, tU/D > 500, the jet branches converge to a quasi-steady state, so we can average over the large period (Fig. 3).To search for the optimal parameters relating to the largest bifurcation angle α, the next runs are performed for different Strouhal numbers at fixed a = 15 Pa leading to V0/U ≈ 0.01 for Re = 500.As observed (Fig. 4), the jet splitting takes place at 0.02 < St < 0.12, the maximum of α is 38º for St = 0.07 (Fig. 3).The value of α is estimated from the curves d1(x) and d2(x) of the typical jet widths determined as the absolute values of z where the time-averaged velocity <u> within the upper and lower branches of the bifurcating free jet reaches the maximum values <u>1max and <u>2max, respectively.
Next, we fix the 'optimal' value of St = 0.07 and vary the amplitude (Fig. 5).At low V0, there is no bifurcation.At smaller St or higher V0, the lower branch is enhanced (as sometimes it is in measurements with one source too [3]) related probably to asymmetric forcing from above.The non-symmetric velocity distributions at lower St and the jet bifurcation at x/D > 10 can be also revealed (Fig. 6) from the transverse profiles of the streamwise mean-velocity component at y = 0, which confirm that St = 0.07 indeed yields the largest jet expansion with the strongest decay of velocity values due to the mixing enhancement.These velocity profiles agree qualitatively with those in the measurements and computations at higher Reynolds numbers (10 3 ≤ Re ≤ 5×10 3 ) having the different types of jet excitations [1,5,[11][12][13].The stronger and earlier decay of the velocity at the axis for lower St or higher a (V0) is also noted (Fig. 7, 8), and far downstream this trend disappears for St < 0.1, a < 10 Pa.Next, a comparative analysis with the jet behaviour under vibrational excitation at Re ≤ 750 investigated in [5,9,10] is performed for the bifurcation plane y = 0 (Fig. 9-16).One can see that mechanical forcing (1)-( 3), which is symmetric along the plane of z = 0, produces always the symmetric jet splitting, unlike to that for the asymmetric acoustic forcing.On the other hand, all other effects are the same, with some shifts in values of St, V0, and x0 (which is the bifurcation origin location).In particular, the x0 values are much lower for the vibrational excitation, acting directly at the jet entrance, whereas the specific area of the acoustic perturbation action may expand from the jet nozzle downstream, up to x ~ 10D, which is a probable reason of the jet splitting delay with the sound source forcing.On the other hand, for the inlet wall vibrations at quite low amplitudes, Z0/D = 0.01 (giving V0/U = 0.0094) and Z0/D = 0.025 (V0/U = 0.0236), the jet splitting starts at x ~ 15D and 10D, respectively (Fig. 10, 12), which looks similar to that for the acoustic forcing (Fig. 2-5).
Another common feature is the mean-velocity decay on the jet axis, demonstrating the earlier and stronger fall for higher forcing amplitudes or lower frequencies at not so large positions of x (Fig. 7, 8, 11, 12).The integral quantity of the jet bifurcation angle α determined from thicknesses d1 and d2 as noted above is presented next for acoustic and vibrational excitations.
The α dependence on the perturbation level (Fig. 14) confirms the previous finding for the α growing with amplitude, and demonstrates also this growth saturation (diminishing effect) and even the small drop of α at large amplitudes, which happens earlier for acoustic forcing.Furthermore, as has been already shown above, the bifurcation angle grows with Strouhal number, reaches the 'optimal' St value, then falls at higher St (Fig. 15).
Generally, someone can find (Fig. 16) that, at least, for Re < 2000, the optimal Strouhal number Stopt grows with Re.This effect can be connected, in particular, with the downstream shift of the bifurcation origin at lower Re, which in turn causes the visible drop of the natural perturbation frequency in a round jet as noticed in [5].Such a downstream shift seen for the acoustic forcing at Re = 500 (Fig. 2-5) may explain, why the corresponding Stopt values in Fig. 15, 16 are smaller than those for inlet vibrations.The considerable drop of Stopt in [4] could also be caused by peculiarities of measurements with the great density change in a helium jet and the ambient air.

Conclusions
The study has analysed the numerical simulation results for a free round jet with transverse excitation of two types: (1) single-mode acoustic forcing from the sound source at the lateral boundary; (2) inlet wall vibrations.Note that the numerical study of the jet bifurcation by the acoustic field has been done for the first time.So a comparison of these results with those [5,9,10] in the presence of vibrational excitation at varied amplitudes, Reynolds and Strouhal numbers produces the unique conclusions.In particular, the optimal value of St is examined, and its dependence on Re is discussed.
It is of interest for future studies to explore the jet behaviour at wider ranges of Re, forcing amplitudes, density drops, for different nozzle shapes and multiple sound sources placed at different planes, combinations of several types of jet excitations.To search the optimal control parameters (to force jet spreading, entrainment rate, mixing efficiency) and to apply the jet flow control methods for heat and mass transfer issues, e.g. for heat exchange enhancement, are the key perspectives too.

Fig. 1 .
Fig. 1.Scheme of the computation domain for a round jet with transverse acoustic forcing: (a) 3D view, (b) side view.