Space-time modulation of turbulence in co-flow jets
Introduction
In this paper, we investigate the response of a moderately turbulent co-flow plane jet subject to a combination of a small-scale random perturbation and a large-scale time-dependent spatially periodic disturbance. We evaluate the dynamic response of the flow in relation to the frequency and length-scale used to stir the turbulent jet flow. Direct Numerical Simulations (DNS) of the Navier-Stokes equations are performed to investigate the role that variation of the frequency and the length-scale associated with the applied spatial modulation have on the downstream developing flow. We focus on the enhancement of mixing properties as a result of variation of frequency and wave-number of the large-scale inflow modulation. Research on modulated turbulence (Von der Heydt, Grossmann, Lohse, 2003, Kuczaj, Geurts, Lohse, 2006) has previously focused on the case of homogenous isotropic turbulence (HIT). Here, our analysis is based on a case of inhomogeneous flow, such as the spatially developing jet, and consider its global response to an external agitation. As will be shown, in the co-flow jet configuration an enhancement of the dissipation rate, ε, can be induced by controlling the frequency and the wave-number of the flow stirring mode. This suggests that time-dependent modulation of turbulence could be used to enhance various properties that are in general often associated with the flow mixing capabilities.
Turbulent flows are characterized by a broad range of length and time-scales. Recent studies suggest the additional existence of preferential frequency modes in turbulence (Von der Heydt, Grossmann, Lohse, 2003, Cekli, Tipton, van de Water, 2010, Kuczaj, Geurts, Lohse, 2006) connected to large-scale modulation. By considering the energy cascade process, these investigations have proposed that such preferential time-scales are intimately associated with the time required for the turbulent kinetic energy, largely present at the large scales of the flow, to cascade down to the small flow scales. Wind tunnel experiments with an active stirring, cycled at preferred frequencies, have shown an enhancement of around 50% in the mean turbulent dissipation rate when the main flow was periodically modulated at these preferred frequencies (Cekli et al., 2010). This suggests that an enhancement in the flow mixing properties could be reached, at given fixed energy input, provided the flow is modulated at the ‘right’ scales. For example, if the dissipation rate could be increased, due to an externally applied disturbance, this would in turn indicate intensified small scales in the flow and an increase in the associated micro-scale mixing. The fact that an increasing in the mixing properties can occur at suitable modulation frequency, offers the possibility to apply the concept of modulated turbulence in generating efficient flow control strategies focusing on mixing applications, e.g., combustion. Despite such investigations, it is also interesting to note that results somewhat opposite to what has been documented so far, namely the enhancement of dissipation and it connection to mixing enhancement, were recently published for a homogenous turbulent flow (Bos and Rubinstein, 2017).
The use of a co-flow jet in this paper relies on its similarity with flow configurations often used in research of premixed combustion of Bunsen-type flames (Vreman, van Oijen, de Goey, Bastiaans, 2009, de Souza, Bastiaans, De Goey, Geurts, 2017). In this paper, we rather concentrates on the general characterization of the response of the co-flow planar jet, e.g. mixing properties, due to the imposed large-scale modulation, without entering in details on the role that the interface dynamics between both stream, as investigated by several groups (Da Silva, Hunt, Eames, Westerweel, 2014, Da Silva, Taveira, 2010, Gaskin, Mckernan, Xue, 2004, Geurts, 2001, Hunt, Eames, Westerweel, 2006, Tordella, Iovieno, 2011, Veeravalli, Warhaft, 1989), has in this context.
By extending the agitation approach used in the reference (Cardoso de Souza, Geurts, Bastiaans, Goey, 2014, de Souza, Bastiaans, De Goey, Geurts, 2017), we investigate the case of a turbulent co-flow jet subject to a space-time modulation pattern. In this paper, by imposing a large-scale external disturbance, that is periodic in space and time, to the turbulent co-flow jet, we address the question whether there are optimum frequencies to stir the flow via inflow perturbations. For instance, such ‘optimum’ frequencies and scales could be beneficial for applications involving turbulent premixed combustion (de Souza, Bastiaans, De Goey, Geurts, 2017, Verbeek, Pos, Stoffels, Geurts, Meer van der, 2012).
The application of a steady modulation via large scale deflection of the flow was shown to lead to a substantially increased response of the co-flow jet configuration (Cardoso de Souza et al., 2014). In that case, it was found that the flow response, characterized for instance in terms of the dissipation rate ε and the jet thickness δz, depends strongly on the imposed length-scales, i.e., the wave-number K. For example, a large response in ε was found near the inflow plane when the flow was disturbed with relatively small length scales, i.e., rather large values of K, while further downstream a response maximum occurs for much larger length-scales of the inflow deflection pattern. Additionally, the width of the mixed region of the jet, δz, was observed to increase faster when the flow was spatially modulated by structures of size comparable to the integral scales of the jet. In general, the results suggest that for such flows ‘resonance’ conditions occur at every length scale of the inflow modulation. At a given modulation wave-number K the largest effect is observed at an associated K-dependent distance from the inflow.
In this paper, we focus on the flow response associated with the introduction of a temporal variation in the amplitude of the large-scale upstream modulation. The response of a flow to an externally imposed disturbance can be characterized in terms of instantaneous flow structures but also by monitoring time-averaged properties of the flow (Cekli, Tipton, van de Water, 2010, Kuczaj, Geurts, Lohse, 2006). Here, we incorporate both types of information and evaluate the response at different modulation frequencies and different perturbations length scales. For instance, the response based on a time-averaged property is evaluated in terms of the dissipation rate, ε, while the instantaneous response of the flow is measured in terms of the total kinetic energy of the flow, ET. In case of ET, we apply a phase-averaging procedure to determine the response during a period of forcing. This analysis is performed at different downstream locations. Results for the conditionally averaged response, ET, show a periodic response amplitude when the main jet is agitated by low modulation frequencies, i.e., ω < ω0, where ω0 is the large-eddy turnover frequency, whereas for higher frequencies, ω > ω0, the response amplitude decays. This is found for any modulation wave-number K, i.e., both at large response amplitudes in case K is sufficiently low and at much reduced amplitudes in case K is large. These results are in accordance with previous investigations on the application of time-modulated turbulence in other flow configurations (Von der Heydt, Grossmann, Lohse, 2003, Cekli, Tipton, van de Water, 2010, Kuczaj, Geurts, Lohse, 2006, Cadot, Titon, Bonn, 2003). Additionally, the effects of the temporal modulation on the flow mixing properties were investigated considering the spreading of a passive scalar injected at the inflow plane. The results show that the mixing of the scalar is affected by the time-dependent spatial modulation. In comparison with a reference unmodulated jet, for example, we observe that a larger scalar surface area tends to be developed when the flow is also modulated in time, specially at frequencies near ω0.
The organization of this paper is as follows. In Section 2, the computational flow model is outlined. The time-dependent modulation strategy and the methods to extract the flow response are also addressed in this section together with an investigation on the sensitivity of relevant flow properties to the grid spatial resolution. Section 3 starts with the results of the unmodulated reference co-flow jet, after which the results of the time-modulated co-flow jet involving a wide range of modulation frequencies are discussed in Section 4. Results of the scalar mixing are discussed in Section 5. Concluding remarks are contained in Section 6.
Section snippets
Layout of the computational model
Firstly, we outline the system of equations and associated boundary conditions, and introduce the computational domain used for the simulations.
The system of Navier-Stokes equations is solved using an incompressible formulation (Cardoso de Souza et al., 2014) yielding the pressure p and the velocity ui in the xi directions as a function of time t:where ρ and ν are, respectively, the density and kinematic viscosity of air at room temperature conditions
Reference unmodulated jet
In this section, we present additional results for the reference unmodulated co-flow jet such as the spatial evolution of the dissipation rate and characteristic flow length-scales. These results provide a reference basis for a comparison with the modulated cases.
Fig. 7 shows a snapshot of the vorticity magnitude on a cross-section yz-plane at x=0, where the evolution and subsequent growth of the inherent instabilities present in the shear layer are directly observed. As can be seen, in the
Periodically modulated co-flow jet
In this section, we present the flow response to the imposed space-time modulation considering a broad range of cases. Firstly, we discuss the phase-averaged results of the response based on the kinetic energy, ET, after which the results of the averaged dissipation rate, ⟨ε⟩xyz, and the modulation effects on the mixing are presented and discussed.
Figs. 9 and 10 show the time dependence of the response signal ET(ω, t) to some selected modulation frequencies for the wave-number at
Mixing Efficiency in time-modulated turbulence
In this section, the effects of the applied modulation on the mixing capability of the co-flow jet are discussed.
Firstly, in Fig. 14 we show snapshots of the surface spreading of the injected scalar for the reference jet and the modulated cases and . In all cases, the snapshots correspond to the scalar instantaneous iso-surface extracted after 5 flow-through times. For a better qualitative impression on the relation between the dissipation rate and the passively spread
Conclusions
In this paper, we applied Direct Numerical Simulations (DNS) to investigate the effects of a time-dependent large-scale modulation on a co-flow planar jet. In these simulations, the jet confined at the inflow region of width D is agitated by a set of specific scales with amplitude varying periodically in space and time. The effects of the imposed time-modulation were assessed based on a parametric study involving a range of modulation frequencies, ω, for two different imposed length scales. The
Declaration of Competing Interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
CRediT authorship contribution statement
T. Cardoso de Souza: Conceptualization, Methodology, Software, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization. R.J.M. Bastiaans: Project administration, Funding acquisition, Resources. L.P.H. De Goey: Project administration. B.J. Geurts: Conceptualization, Writing - original draft, Writing - review & editing, Supervision, Funding acquisition, Resources.
Acknowledgements
The authors would like to acknowledge the Dutch Technology Foundation STW for the financial support, The Netherlands Computing Facilities (NCF) for grant: SH-061, and the computing facilities of the High Performance Computing Center-NPAD of the Federal University of Rio Grande do Norte - UFRN.
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