Toward the use of a reduced‐order and stochastic turbulence model for assessment of far‐field sound radiation: Low Mach number jet flows

This work presents a novel framework for the evaluation of far‐field sound radiation, specializing to a low Mach number jet flow. The framework comprises an analytical and a numerical part. In the analytical part, a low Mach number asymptotic analysis is presented to obtain the spectral sources of sound radiation starting from a pressure wave equation that is directly obtained from the Navier–Stokes equations. The derivation procedure is based on the fundamental premise of Lighthill's acoustic analogy. In the numerical part, a reduced‐order model for turbulent flow, the one‐dimensional turbulence model, is used to simulate the velocity field of a low Mach number turbulent round jet, assumed fully developed and statistically steady, hence homogeneous in the azimuthal direction and for a cross‐sectional slab assumed locally homogeneous in axial direction, too. The generated velocity field is used for the calculation of the spectral pressure sources, and consequently, to estimate the far‐field sound pressure level (SPL) of the jet. The results of the analysis are compared against available experimental SPL measurements from a subsonic jet. A reasonable agreement is obtained for the SPL. Furthermore, the analysis sheds light into the different contributions of the subsonic flow to sound radiation. The framework is readily extendable for any kind of low Mach number flow, including variable density flows and jet flames.


INTRODUCTION
Turbulence is an important source of sound radiation.The first efforts to relate sound radiation with turbulence can be traced back to Lighthill and the formulation of his acoustic analogy [1].It is somehow surprising to see that, up to this day, several state-of-the-art investigations still profess that an understanding of the sources of sound radiation from first principles has not being achieved.This has led to the appearance of several phenomenological descriptions for explaining turbulent noise, being the large-scale (LS) and fine-scale similarity theory probably the most widely tested and validated of these descriptions, see [2,3].Phenomenological models, however, cannot be easily generalized.Lighthill's basic premise for the acoustic analogy, in that sense, was not a phenomenological approach, but one starting from first principles.The fundamental premise was to estimate the sound radiated from a given fluctuating fluid flow, by rearranging the fluid flow equations, in a way in which they could be rewritten as the equations of the propagation of sound, that is, a wave equation, in a uniform medium at rest [1].The medium at rest is preferred, given that this configuration avoids the concern of feedback or coupling of the sound on the flow, as well as concerns regarding modifications of the radiated sound due to convection with turbulent flow and propagation by it at variable speed [1].In other words, Lighthill's original idea revolves around the treatment of the pressure as an independent variable from the velocity field, and as such, pressure cannot be transported by the velocity field.There have been several attempts to obtain the generalized wave equation with sources for sound radiation.We cite remarkable work, for example, from Lilley [4], which derives a generalized third-order nonlinear pressure wave equation.In [5], Lilley's equation is linearized by shifting nonlinearities of the pressure transport as (highly complicated) source terms, some of which are later neglected in a specialized treatment for mixing layers [5].Lilley also proposes a novel generalized nonlinear wave equation utilizing a decomposition of flow variables into a linear and a nonlinear part, similar to a Reynolds decomposition [6].Other commendable efforts are made by Goldstein, who proposed a general set of exact linearized inhomogeneous Euler equations, which framed several phenomenological noise prediction methods in a first-principle basis [7].Although quite general, Goldstein's generalization in [7] also returns a nonlinear pressure wave equation, given that the choice of a base flow, for example, the steady mean flow of a jet, leads to an inhomogeneous wave equation for a moving medium.Overall, all methods demonstrate tremendous analytical progress and extend Lighthill's analogy, yet fall short in the unambiguous identification of sound sources due to the required linearization step.Only when the sources of sound radiation are identified in an exact way, could then accurate models be formulated that should be capable of optimizing very small deviations in strict regulated sound pressure levels (SPLs), for example, ±3 dB, as commented in [8].
To address the shortcomings, Hu et al. [9] proposed an acoustic analogy specialized for turbulent channel flows, which derives a pressure wave equation utilizing Mach number asymptotics [10,11].We note that by using Mach number asymptotics, it is possible to return the so-called incompressible flow equations, starting from the generalized Navier-Stokes equations.The work in [9] could have been the much needed closure for the incomplete version of Lighthill's acoustic analogy; nonetheless, there was a major inconvenience.Indeed, the monopole term obtained in [9] for the pressure wave equation was a source term depending itself on the very same pressure.To circumvent this issue, the monopole term was related to a viscous dissipation function utilizing a very loose order of magnitude analysis.Although such an order of magnitude analysis is common in the Mach asymptotics framework, it lacked consistency in the work of [9].This small but crucial step prevents the generalization of the approach in [9].
In this work, we present what we believe is a more consistent Mach asymptotic analysis applied to a very general form of the pressure wave equation.The framework, however, requires facing the usual difficulties associated with direct numerical simulations (DNSs) in order to be of any use, given that extensive space-time data must be mapped onto the spectral domain in order to obtain the power spectral density of the pressure, and subsequently, SPLs.To avoid the utilization of DNSs, we verify the results of our analytical Mach asymptotic procedure utilizing a reduced-order stochastic turbulence model, the one-dimensional turbulence (ODT) model [12].ODT is a turbulence model that is ideal for unsteady turbulent flows that can be statistically characterized by only one dominant direction of flow gradients.This significantly reduces the computational effort for the generation of far-field acoustic pressure spectra, because an otherwise time-to-frequency and 3-D physical space-to-wave number Fourier transform reduces to a time-to-frequency and 1-D Fourier transform.
The structure of this contribution is as follows.First, we present the derivation of the utilized pressure wave equation and its corresponding Mach number asymptotic analysis.Afterward, we present a very general overview of the ODT model.The ODT model is used to estimate the far-field acoustic SPLs of axisymmetric low Mach number turbulent jet flows.We omit several details of model implementation, referring instead the interested readers to [13].We then present the flow configuration and initial and boundary conditions used for evaluating the suggested framework, and we compare numerical results of the SPL to available experimental data [14].Finally, we give some closing comments and conclusions.

GENERALIZED PRESSURE WAVE EQUATION AND LOW MACH NUMBER ASYMPTOTICS FOR ISOTHERMAL TURBULENT JET FLOWS
To derive the generalized form of the pressure wave equation that will be used in this work, we start by considering the continuity, momentum and energy equations in a general form.The form at consideration utilizes constant transport coefficients (dynamic viscosity , thermal conductivity  th , and specific heat capacities, e.g., at constant pressure   ), and a Newtonian total stress tensor of the form Here,  is the pressure,  is an identity matrix,  is the second viscosity coefficient (not to confuse with the bulk viscosity, and  ≠ ), also assumed constant, ∇ is the usual Nabla vector operator acting on a coordinate system determined by a generic position vector , and  is the velocity vector field, whereas ∇ ⋅  denotes the divergence of the velocity field with the usual scalar product.Also,  = (1∕2)[∇ + (∇)  ] is the symmetric part of the velocity gradient contributing to the shear stress.The governing equations are valid for ideal gases of fixed composition, such that In Equation ( 2),  gas is the specific gas constant,  is the density, and  is the temperature.We utilize the following nondimensional scalings, for example, in terms of the bulk-flow variables, denoted with a subindex , and a bulk-flow length scale , for example, the jet diameter.We purportedly introduce an arbitrary frequency scale  for the time variable , and we scale  in terms of All nondimensional quantities are denoted with * .Note that Φ = 2 ∶ ∇ is the kinetic energy dissipation rate, where ∶ indicates a double-dot product (or double scalar product) and ∇ is the gradient of the velocity field.The pressure reference scale is also given by the ideal gas speed of sound  , , that is,   =    2 , ∕.It is also related to   and   by the ideal gas law   =   ∕( air   ).This set of scales allows the formulation of the Mach, Reynolds, Prandtl, and Strouhal numbers, as follows: The nondimensional form of the governing equations at consideration is then []  (  *  * ) [] *  *  * +  * ( []  *  * + We comment briefly on these equations.Equations ( 5) and ( 6) are the usual continuity and momentum equations in a strong differential form.Note that • denotes a dyadic product.Equation ( 7) is a temperature equation that can be derived in the usual form, starting from the total energy equation (in strong differential form), and then substituting the (weak) equation for specific kinetic energy,   = (1∕2)( ⋅ ), utilizing the definition   = ℎ − ∕ +   , where   is the total specific energy and ℎ is the specific enthalpy, related to  by the specific heat capacity, that is, dℎ =   d.The latter enthalpy definition requires then the use of a weak differential form for the temperature equation.In addition to the temperature equation, Equation ( 8) is a pressure equation, which is obtained by combining the continuity equation in the form D∕D = −∇ ⋅  with the total differential form of the equation of state, and relating later this result to the temperature equation.The system of equations given by Equations ( 5)-( 8) is then our closed system of equations of choice for this work.It is possible to take the divergence of Equation ( 6), multiply this result by the local speed of sound   (in nondimensional terms), and subtract this from the partial derivative of Equation ( 8) with respect to time, in order to obtain a nonlinear wave equation for the pressure.Such nonlinear wave equation has the advantage that, as Lighthill intended, the pressure is a fully independent variable, and the only nonlinear term is due to the convective transport of the pressure by the velocity field, We consider low Mach number jets in this contribution, that is, the limit [] → 0. In the following, a decomposition of the flow variables ( * ,  * ,  * and  * ) in terms of a power series of the Mach number is performed, following the formal framework of Mach asymptotics described in [10,11].As an example for the pressure, this decomposition is This is an infinite series in terms of [].We evaluate two distinctive cases for the Strouhal number in Equations ( 5)-( 9).The first case is given by a frequency scale comparable to the inverse of the acoustic time scale,  =  , ∕, such that The second case is for any other arbitrary choice of , corresponding to small or large time scales of the flow, [] ≠ [] −1 , for example, a turbulence time scale.Substituting the Mach power series for each flow variable in Equations ( 5)-( 9), and later equating coefficients of Mach powers of the same order, it is possible to arrive at governing equations that are valid at different orders of accuracy (  ),  ≥ 0 being an arbitrary integer exponent.For brevity, we focus on the Mach asymptotic results solely for Equation (9), which utilize the conclusions from the asymptotics of Equations ( 5)- (8).We also shift the discussion of the case [] ≠ [] −1 elsewhere, to a future publication.We solely stress that the asymptotic forms of the wave equation derived for the case [] ≠ [] −1 lead to a pressure constancy in time and space for  * 0 , spatial uniformity for  * 1 (with some time dependency  * 1 ( * )), and a Poisson equation for  * 2 , which is the well-known equation for enforcing the zero divergence condition in low Mach number flow.None of these equations correspond in the strict sense to wave equations, which is the reason why the case [] = [] −1 is of larger interest.Selecting [] = [] −1 in the asymptotic analysis, the zeroth-, first-, and second-order contributions in power of Mach yield the equations These are insightful equations because, when looking at the dynamics established by the acoustic time scale, we can then guarantee that the pressure behaves, up to order ( 0 ) as a constant, up to order ( 1 ) as an acoustic wave in a uniform medium at rest without sources (other than some initial or boundary perturbation), and up to order ( 2 ) as an acoustic wave in a uniform medium at rest with sound radiation sources.In fact, the sources are identifiable as effects corresponding to heat conduction, turbulent heat flux, the so-called fluctuating Reynolds stress [1], and the rate of change of the flow kinetic energy.On one hand, the time rate of change of   is a monopole source, and it also becomes an additional source for sound radiation when the smaller effect due to the dissipation of kinetic energy per molecular heat transfer is considered.Turbulence, on the other hand, seems to generate sound radiation both due to the fluctuating Reynolds stress term and the turbulent heat flux.We will now derive the SPL associated with the newly found wave equations.To that extent, the first step is to apply a time-to-frequency Fourier transform (CTFT) on Equation (13), such that a Helmholtz equation for  * 2 is obtained.It is noted that although the wave equation for  * 1 has no sources,  * 1 itself is governed by a transport equation that grants it a dynamic character.To that extent, the CTFT can also be applied on that transport equation for  * 1 .Most of the procedure for the derivation of the SPL is omitted here for brevity.We note, however, that the procedure mostly resembles that applied in [9], and the key issue summarizes to the application of a Green's function approach combined with a far-field approximation that originates the appearance of 3-D discrete spaceto-wave number Fourier transforms (DSFT).To that extent, the spectral form of p * 2 , as a function of both an acoustic measurement coordinate in the far-field  * , and the angular frequency Here,   denotes a CTFT and  −1 3−D an inverse DSFT, which are defined in a nonunitary way as usual.We note that all results from the DSFTs are evaluated at the acoustic wave number  =  * ∕ √  * 0 .Also,  is a 1-vector of three components and  is a square 1-matrix of nine components.The spectral form of p * 1 is not obtained from the wave equation given the absence of sources, but either from the first-order asymptotic pressure transport equation, or from the first-order momentum equation, Equations ( 14)-( 15) are striking, given that in spectral terms, the pressure is entirely determined by the velocity field  * 0 , considering that  * 0 and  * 0 are constants in the case of isothermal jets.The total spectral pressure is the sum of p * 1 and p * 2 at an equivalent order of accuracy in terms of Mach powers.Given that the wave equation at () has no sound radiation sources, the correct order of accuracy to estimate the sound radiation, and thus, the order of accuracy at which the summation of p * 1 and p * 2 occurs is ( 2 ).As such, the result of p * 1 needs to be multiplied by [𝑀𝑎].Note also that due to CTFT properties, considering the definition of the nondimensional ordinary frequency as  * =  * ∕(2), we have p * ( * ) = (2) −1 p * ( * ), where −∞ <  * < ∞.Therefore, the total spectral pressure responsible for sound radiation is The SPL is defined by the power spectral density of p * , namely,  *  = ⟨( p * ) † p * ⟩, where ( p * ) † is the complex conjugate of p * and the angle brackets ⟨⟩ denote an ensemble average.Consequently, where  ref = 20 Pa.Note that the spectrum  *  must be scaled according to the overall SPL (OSPL)  *  ( * , 0), which is not a result of the procedure so far, and must be provided as an input.We find a method to relate the OSPL to the pressure difference Δ ′ corresponding to underexpanded subsonic jets discharging to the atmosphere using classical gas dynamics.The method was found by analogy to the distinction made between the local design Mach number at the nozzle exit, []  , and the fully expanded jet Mach number []  in supersonic jets [15].For brevity, we do not detail the method, but we note that such Δ ′ can be estimated for some subsonic jets in [14], given that both []  and []  were measured, whereas

CALCULATION OF THE SOURCES OF SOUND RADIATION WITH THE ODT MODEL
So far, no modeling has been done for the sound radiation sources.To avoid the use of DNSs, we utilize the ODT model for the evaluation of the sources in Equations ( 14) and (15).To that extent, the only modeling required is to recall that ODT is a 1-D model, which nonetheless allows full length-and time-scale resolution.ODT calculates all velocity components resolved in the 1-D domain [12,13], and allows calculation of all components of the fluctuating Reynolds stress, as well as the kinetic energy, all present in Equations ( 14) and (15).Thus, our only modeling is that we reduce the complexity of the  −1 3−D operators to 1-D, that is,  −1 1−D .On the assumption that small-scale (SS) processes are able to represent the collective behavior of turbulent flow, ODT focuses on a simplified representation of scalar advection by SS turbulence, that is, the modeling of nonlinear advection by means of Lagrangian mappings of fluid parcels.The mapping of choice is a triplet map [12], and a long enough sequence of mappings represents the effects of turbulent eddies in the statistical ensemble limit.Although originally applied for solenoidal flows (so-called incompressible flows) with statistical 1-D flow properties [16], there have been extensions of the ODT model to include variable density effects [17,18].Here, we comment on the desired isothermal turbulent round jet application.According to Equations ( 14)-( 15), we only require the velocity field  * 0 in order to estimate the sources for sound radiation.As a result of the asymptotic analysis, although not discussed so far,  * 0  * 0 ∇ * ⋅  * 0 =  * 1 ∕ * , and  * 0  * 0 ∕ * = −∇ *  * 1 , leading both to Equation (15).This implies that  * 1 has a role on the enforcement of the divergence condition, and on the rate of change of  * 0 in time, both things being equivalent when [] = [] −1 .This is a momentum redistribution mechanism among velocity components, strikingly similar to the kinetic energy redistribution in solenoidal flows due to pressure scrambling.In ODT, the pressure is not a flow variable, and instead, its effects on turbulent transport are only modeled.Indeed, the pressure scrambling effect is modeled by a kernel function [16].The wave equation treatment with sources for the determination of  is, in that sense, compatible with the ODT rationale.Furthermore, we make the modeling choice of assuming that the effect of  * 1 is already considered in ODT by the kernel mechanism, and we simulate in ODT the otherwise isothermal, constant properties solenoidal flow where ∇ * ⋅  * 0 = 0. To that extent, the ODT model in a vector formulation suffices [16].Due to the brevity of this presentation, we do not comment further on the ODT model.Instead, the reader is referred to [13] for further details on the application to turbulent round jets.
Figure 1 shows the flow configuration sketch, detailing the position of the microphone arrays in the experimental setup from [14].The position of the ODT domain in the streamwise coordinate of the jet is arbitrary, given that the line only sees the temporal change of the flow along the radial coordinate.However, an assumption on streamwise and azimuthal statistical homogeneity is required, which implies that the position of the ODT domain corresponds at least to a zone in which the jet is self-similar.Table 1 details the simulation inputs.Figure 2 shows a space-time visualization of the nondimensional kinetic energy of the flow for one ODT simulation.An ensemble average of flow simulations is shown in Figure 3.This is, in fact, the rationale behind ODT, in the sense that Figure 2 does not yield the continuum flow statistics, but Figure 3 does.This is also better represented for the evaluation of the SPL as seen in Figure 4, where the ensemble average is required in order to produce a smooth spectrum.Note that the ensemble average is demanded by Equation (17). Figure 5(a) shows the obtained SPL in the one-third octave band.As a reference, the experimental measurement data [14], as well as the LS and SS similarity spectra are shown [2].Finally, Figure 5(b) shows the contributions of order () and TA B L E 1 Simulation inputs following experimental conditions [14].

Simulation parameter Value
[]  (Design Mach number, see [15]) 0.41 []  = [] (Jet Mach number) 0.4 [] (Jet Reynolds number based on the jet diameter) ≈ 553 × 10 3 [] (Jet Prandtl number, air) 0.71  sideline (Sideline distance to measurement, see Figure 1  ( 2 ) to the SPL (the spectra are now shown in the fully resolved narrowband representation).The agreement between the calculated SPL and the measured SPL is striking.Although  * 1 dominates for most of the frequency range,  * 2 is the dominant source of sound at high frequencies.

CONCLUSIONS
We have presented a novel analytical and numerical framework for the estimation of far-field acoustic SPLs in turbulent jet flows.The analytical framework utilizes Mach number asymptotics in order to derive a linear pressure wave equation with easily identifiable sources for sound radiation, as in Lighthill's acoustic analogy.The numerical framework utilizes a reduced-order, full-scale resolving, stochastic turbulence model, ODT, in order to estimate the sources for the wave equation.Overall, the preliminary results show that the proposed framework is capable of reasonably predicting the SPL, although further validation is pending for future work.

A C K N O W L E D G M E N T S
This work was supported by the BTU Graduate Research School (Conference Travel Grant).
Open access funding enabled and organized by Projekt DEAL.

F I G U R E 5
(a) Obtained SPL in one-third octave band.(b) The () and ( 2 ) contributions to SPL.