(2020) Knudsen Number Effects on the Nonlinear Acoustic Spectral Energy

PHYSICAL REVIEW E 101, 023101 (2020)

Knudsen number effects on the nonlinear acoustic spectral energy cascade

Shubham Thirani,* Prateek Gupta, and Carlo Scalo School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47906, USA

(Received 18 September 2019; accepted 2 January 2020; published 3 February 2020)

We present a numerical investigation of the effects of gas rarefaction on the energy dynamics of resonating planar nonlinear acoustic waves. The problem setup is a gas-filled, adiabatic tube, excited from one end by a piston oscillating at the fundamental resonant frequency of the tube and closed at the other end; nonlinear wave steepening occurs until a limit cycle is reached, resulting in shock formation for sufficiently high densities. The Knudsen number, defined here as the ratio of the characteristic molecular collision timescale to the resonance period, is varied in the range Kn = 10−1–10−5, from rarefied to dense regime, by changing the base density of the gas. The working fluid is Argon. A numerical solution of the Boltzmann equation, closed with the Bhatnagar- Gross-Krook model, is used to simulate cases for Kn 0.01. The fully compressible one-dimensional Navier- Stokes equations are used for Kn < 0.01 with adaptive mesh refinement to resolve the resonating weak shocks, reaching wave Mach numbers up to 1.01. Nonlinear wave steepening and shock formation are associated with spectral broadening of the acoustic energy in the wavenumber-frequency domain; the latter is defined based on the exact energy corollary for second-order nonlinear acoustics derived by Gupta and Scalo [Phys.Rev.E 98, 033117 (2018)], representing the Lyapunov function of the system. At the limit cycle, the acoustic energy spectra exhibit an equilibrium energy cascade with a −2 slope in the inertial range, also observed in freely decaying nonlinear acoustic waves by the same authors. In the present system, energy is introduced externally via a piston at low wavenumbers or frequencies and balanced by thermoviscous dissipation at high wavenumbers or frequencies, responsible for the base temperature increase in the system. The thermoviscous dissipation rate is shown to scale as Kn2 for fixed Reynolds number based on the maximum velocity amplitude, i.e., increasing with the degree of flow rarefaction; consistently, the smallest length scale of the steepened waves at the limit cycle, corresponding to the thickness of the shock (when present) also increases with Kn. For a given fixed piston velocity amplitude, the bandwidth of the inertial range of the spectral energy cascade decreases with increasing Knudsen numbers, resulting in a reduced resonant response of the system. By exploiting dimensionless scaling laws borrowed by Kolmogorov’s theory of hydrodynamic turbulence, it is shown that an inertial range for spectral > energy transfer can be expected for acoustic Reynolds numbers ReUmax 100, based on the maximum acoustic velocity amplitude in the domain.

DOI: 10.1103/PhysRevE.101.023101

I. INTRODUCTION and timescales (higher wavenumber and frequency content), the thermoviscous dissipation increases, causing the total Nonlinear acoustic processes are observed in various en- perturbation energy to decrease in time [Fig. 1(b)] but also gineering applications such as harmonic resonators [1,2], limiting the spectral extent of the inertial cascade range [e.g., thermoacoustic engines [3,4], ultrasonic imaging [5,6], and to k 104 at time t = t ,inFig.1(c)]. high-speed boundary layers [7,8]. An important feature of 3 For a resonant system with continuous external energy nonlinear acoustic processes is wave steepening, typically injection driven to its nonlinear limit cycle, such as the one associated with thermodynamic nonlinearities [9]: regions considered in the present manuscript (Fig. 2), an equilibrium of positive pressure fluctuations are characterized by higher interscale energy transfer is achieved in the spectral space: the temperatures and, hence, locally higher sound speeds (and resonant energy cascades down to higher harmonics reach- vice versa); this leads to the enhancement of spatial gradients ing progressively smaller length scales, until thermoviscous in the wave propagation speed yielding wave steepening [10]. dissipation takes over, sustaining the shock thickness. The In the spectral space, wave steepening entails energy cascade bandwidth of the thus-obtained broadened energy spectrum into higher harmonics, associated with the formation of pro- denotes the degree of wave steepening attained by the sys- gressively smaller length scales and timescales [11]. Figure 1 tem and it is inversely related to the shock thickness. Such illustrates such an energy cascade by showing the temporal resonant energy injection can be achieved via fluid dynamic evolution of average velocity magnitude and wavenumber instabilities or externally applied forcing. In thermoacoustic spectral density of a freely propagating nonlinear wave in a devices for example, the energy injection takes place due to periodic domain. Due to generation of smaller length scales thermoacoustic instabilities. The presence of nonlinear wave effects in thermoacoustics have been identified and studied experimentally [12,13] and numerically [4]. More recently, *[email protected] Gupta et al. [4] demonstrated the existence of an equilibrium

FIG. 1. Velocity perturbation in a high amplitude acoustic traveling wave (a), evolution of normalized spatial average of u2 (b), and velocity 2 spectra |uˆk | (c) at times t0, t1, t2, t3. The spectral broadening occurs due to the nonlinear terms in the governing equations resulting in the energy cascade from larger to smaller length scales. spectrum, resulting from a balance between the thermoa- physical length scale, quantifies the degree of rarefaction of coustic energy production and thermoviscous dissipation at a gas. For a single-frequency acoustic wave, the relevant length scales corresponding to the shock thickness. Similar length scale is taken as the acoustic wavelength. That is, observations have been made experimentally for a gas in a even at standard atmospheric conditions, a high frequency resonance tube periodically driven by a piston oscillating at wave, say 260 MHz [28], yields Knudsen numbers close to frequencies close to the fundamental frequency [2,14,15]. 0.02 and is, hence, in the rarefied regime. Based on this no- The aforementioned studies only focus on acoustic nonlin- tion, experiments were conducted [20,21] focusing on noble earities in dense gases, in most cases air at standard temper- gases only excited by a high frequency pulse (∼11 MHz) ature and pressure (STP). However, nonlinear acoustic wave in a transmitter-receiver system to investigate acoustic wave propagation also plays a crucial role in controlling transition propagation in a rarefied medium, with Knudsen numbers in to turbulence in high-speed boundary layers, which are often the range 0.002–2. It was found that the attenuation rate of the in rarefied flow conditions. In canonical cases, transition to sound wave increased with the Knudsen number, due to so- turbulence is in fact governed by the amplification of ultra- called translational dispersion: In rarefied environments, only sonic waves with energy mostly confined in the boundary the faster moving molecules contribute to the macroscopic layer [16–19], acting as an acoustic wave guide. For such wave transmission mechanism due to larger mean free path conditions, the flow can be considered in the rarefied regime values, thus accelerating the high-frequency waves leading merely due to the very high frequencies involved, and even to wave dispersion and, hence, attenuation [28]. Inspired by more so for high-altitude flight with characteristic pressures these experimental attempts, we herein focus on a canonical and densities of the order of 1 kPa and 0.01kg/m3, respec- flow setup; a one-dimensional system excited at its resonant tively. This consideration motivates the current work, in which frequency with Argon as the working fluid. we explore the effects of gas rarefaction on the behavior of Other theoretical investigations [24–27] confirm the in- resonant nonlinear acoustic waves. crease in the attenuation of the sound wave with increas- Propagation of sound waves in rarefied media has been ing Knudsen number, also showing how Navier-Stokes-based analyzed in great detail both experimentally [20–23] and solutions deviate from the experimental and kinetic theory numerically [24–27]. The Knudsen number, Kn, defined as results for high Knudsen numbers. This is not surprising, the ratio of the molecular mean free path to the relevant as the Navier-Stokes equations are based on a continuum

023101-2 KNUDSEN NUMBER EFFECTS ON THE NONLINEAR … PHYSICAL REVIEW E 101, 023101 (2020) hypothesis and are hence not valid for high Knudsen numbers, temperature of the gas. The walls of the tube and piston are where the Boltzmann equation is better suited. In this work, in considered adiabatic and the system is modeled only in one fact, we have used the Navier-Stokes equations for cases with dimension, x, with total length, L = 1 m. The gas inside the Kn < 0.01 and the Boltzmann equation for Kn 0.01. The tube is Argon. novelty of the present contribution lies in extending the degree We decompose the instantaneous pressure p, velocity u, of fidelity of the Navier-Stokes simulations to an advanced density ρ, and temperature T ,as (weak-)shock-resolving numerical framework, as well as a = τ + , , = + , , systematic spanning of the Knudsen and acoustic Reynolds p p0( ) p (x t ) u u0 u (x t ) number space; the effects of the latter, in particular, have ρ = ρ0 + ρ (x, t ), T = T0(τ ) + T (x, t ), (3) not been considered in previously. The approach adopted , ,ρ here in fact relies on the theoretical framework by Gupta where p u , and T are the corresponding fluctuating and Scalo [11] on the nonlinear acoustic spectral energy quantities. Here, x and t refer to the spatial and temporal coordinates respectively. The base state for velocity is zero, cascade, which stresses the relative importance of nonlinear = ρ acoustic effects over thermoviscous dissipation, measured by u0 0. Since the domain is closed, the base density, 0,is constant, and as discussed later, the base pressure, p0(τ ), and the acoustic Reynolds number. τ τ The paper is organized as follows. A description of the temperature, T0( ), increase following a slow timescale, ,as problem setup and the parameter space investigated is found p0(x, t ) = p0(t ) p0(τ ), T0(x, t ) = T0(t ) T0(τ ), (4) in Sec. II. Details of the continuum gas dynamics equations and the associated numerical scheme are presented in Sec. III, due to thermoviscous dissipation. All base-state quantities are followed in Sec. IV by a description of the Bhatnagar-Gross- spatially uniform. Krook (BGK) model and the numerical scheme employed to As mentioned earlier, the Knudsen number is the ratio of solve it. The results from the simulations are presented in the mean free path of the gas molecules to the characteristic Sec. V, with subsections focused on comparison between the (macroscopic) length scale of the problem; in the present Navier-Stokes and BGK solutions, steepening of the planar context it can be written as the ratio of two timescales [29], acoustic wave and resonance amplitudes, fluctuation intensity profiles, the equilibrium spatial and temporal spectra, and the 1 ωres μ πKBT0 budgets for the base internal energy. We have also introduced Kn = √ = , (5) 2πγ ν p L 2m scaling parameters to collapse the acoustic energy spectra 0 in the wavenumber-frequency domain for different Knudsen where μ is the dynamic viscosity, ν is the collision frequency, numbers and provide an analytical estimate of the smallest γ = 1.6702 is the ratio of specific heats for Argon, KB = length scale which is of the same order as shock-thickness. 1.38 × 10−23 m2kg/s2K is the Boltzmann constant and m = 66.3 × 10−27 kg is the molecular mass of Argon. Based on II. PROBLEM SETUP the piston velocity amplitude, we define the piston Reynolds

The problem setup, shown in Fig. 2, consists of a closed number, Reup ,as, tube with a piston at one end, oscillating at the tube’s funda- ρ U , L mental resonant frequency ω , calculated as Re = 0 p max . (6) res up μ ω = πa /L, (1) res 0 Later we deﬁne another (more dynamically relevant) and with instantaneous velocity, Reynolds number based on the maximum velocity amplitude achieved in the domain, Umax [Eq. (26)]. u (t ) = U , sin(ω t ), (2) p p max res We have spanned ﬁve orders of magnitude for Kn and three where Up,max is the piston velocity amplitude. The adiabatic orders of magnitude for Reup in our simulations (see Table I). speed of sound, a0, is calculated at the initial base-state As the Knudsen number approaches unity, the mean free path theoretically approaches the length of the domain, hence the base state inside the tube approaches near-vacuum conditions. To avoid such conditions, all the Kn values considered are below unity. The initial value for the base-state temperature of the gas is 300 K for all the cases and the base-state values for density and pressure are evaluated accordingly for each Knudsen number using Eq. (5). The initial base-state values used to achieve various Knudsen numbers are reported in Table II.

III. FULLY COMPRESSIBLE ONE-DIMENSIONAL NAVIER STOKES EQUATIONS

FIG. 2. Illustration of the investigated one-dimensional piston- In this section, we present the fully compressible one- tube setup during maximum compression of the gas at the adi- dimensional Navier Stokes equations in Sec. III A followed ◦ batic wall (t1) and approximately 180 later in the shock resonance by a description of the numerical scheme used to solve them cycle (t2). in Sec. III B.

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TABLE I. Piston velocity amplitudes, Up,max, for the simulation parameter space considered.

←− Dense Rareﬁed −→ −5 −4 −3 −2 −1 Up,max Kn = 10 Kn = 10 Kn = 10 Kn = 10 Kn = 10 = . × −4 / × −3 / / / / Reup 0 2410 m s410 m s 0.04 m s 0.4 m s4ms = . × −5 / × −4 / × −3 / / / Reup 0 02 4 10 m s410 m s410 m s 0.04 m s 0.4 m s = . × −6 / × −5 / × −4 / × −3 / / Reup 0 002 4 10 m s410 m s410 m s410 m s 0.04 m s

A. Mathematical model used the second-order Runge-Kutta scheme with a constant The fully compressible one-dimensional Navier Stokes CFL of 0.5 for integrating the Navier-Stokes Eqs. (7)–(9) equations comprise the conservation equations for mass, mo- and (11) in time utilizing the staggered spectral difference mentum, and energy for the system. They read (SD) spatial discretization approach introduced by Kopriva and Kolias [31]. In the SD approach, the domain is discretized ∂ρ ∂ ρ + ( u) = , into mesh elements and within each element, the Lagrange ∂ ∂ 0 (7) t x polynomial reconstruction of variables allows numerical dif- ∂ ∂ ∂ p ∂ 4 ∂u ferentiation with spectral accuracy. (ρu) + (ρu2 ) =− + μ , (8) ∂t ∂x ∂x ∂x 3 ∂x To accurately resolve all length scales in the flow we have combined the SD approach with the AMR approach as first ∂ ∂ ∂ 4 ∂u ∂q (ρE ) + [u(ρE + p)] = u μ − x , (9) introduced by Mavriplis [32]. The AMR approach allows ∂t ∂x ∂x 3 ∂x ∂x us to focus computational resources to adequately resolve unsteady shock waves. To this end, we expand the values of a where ρ, q , and E are the instantaneous density, heat flux in x generic variable φ local to the cell in the Legendre polynomial x direction, and total energy per unit mass, respectively. The space as total energy per unit mass, E, is defined as = + 1 2, E CvT 2 u (10) N φ = φ˜iψi(x), (13) where Cv is the specific heat capacity at constant volume. Since only a monatomic gas (Argon) is considered in this i=1 work, bulk viscosity effects are neglected. The conservation equations are closed by the ideal gas equation of state, where ψi(x) is the Legendre polynomial of (i − 1)th degree. The polynomial coefficients, φ˜i, are utilized for estimating p = ρRT, (11) the local resolution error, ε, defined as [32] where T is the instantaneous temperature and R = / 208.132 J/kg K is the specific gas constant. The dynamic vis- φ˜2 ∞ 2 1 2 2 N 2 fε (n) −σ n r ε = + dn , fε n = ce , cosity is calculated using the power law, μ = μ(T/T ) , where + + ( ) μ 2N 1 N+1 2n 1 , T , and r are the reference dynamic viscosity, reference (14) temperature, and the exponent for the power law, respectively. The values for μ, T , and r are [30] where fε is the exponential fit through the coefficients of the −5 μ = 2.117 × 10 Pa s, T = 273.15 K, r = 0.81. (12) last four modes in the Legendre polynomial space. As the ε Please note that such reference values do not correspond to estimated resolution error, , exceeds a predefined tolerance, the ones used to define the initial base state in Table II. the cell divides into two subcells, which are connected utilizing a binary tree. The subcells merge together if the resolution error decreases below a predefined limit. The B. Numerical scheme numerical fluxes at cell interfaces were evaluated using the We have performed shock-resolved direct numerical simu- Harten-Lax-van Leer-contact (HLLC) Riemann solver [33]. A lations (DNS), up to Mach number 1.01 of the compressible schematic of the spatial discretization scheme, the Lagrange 1D Navier-Stokes Eqs. (7)–(9) and (11) with adaptive mesh polynomial reconstruction and the numerical flux interfaces, refinement (AMR) [11] for cases up to Kn = 10−3.Wehave as used by Gupta and Scalo [11], is provided in Fig. 3.

TABLE II. Tabulation of initial base state values of density, pressure, and temperature for the span of Knudsen numbers considered, where −2 3 ρref = 1.1456 × 10 kg/m , pref = 715.332 Pa, and Tref = 300 K.

←− Dense Rareﬁed −→ Initial base state values Kn = 10−5 Kn = 10−4 Kn = 10−3 Kn = 10−2 Kn = 10−1

−1 −2 −3 −4 p0 pref 10 pref 10 pref 10 pref 10 pref −1 −2 −3 −4 ρ0 ρref 10 ρref 10 ρref 10 ρref 10 ρref T0 Tref Tref Tref Tref Tref

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FIG. 4. Schematic of the staggered ﬂux conservative spatial discretization used for the BGK solver depicting the ﬂux faces (i+1/2) and cell centers (i).

= , , FIG. 3. Schematic of (a) the polynomial reconstruction of a where u (u v w) is the macroscopic flow velocity. Other quantity φ(x) over a one-dimensional domain consisting of three macroscopic parameters of the flow are calculated from mo- elements, and (b) the staggered spectral difference spatial discretiza- ments of the density distribution function: tion approach used with solution (•) and flux points (|)withina single element. Ncv is the number of elements and p is the number ρ = f (c )dc ,ρu = cf (c )dc , of solution points within an element. c ∈R3 c ∈R3 1 2 IV. THE BOLTZMANN EQUATION T = |c − u | f (c )dc . (19) 3ρR c ∈R3 To study the effects of rarefaction on planar nonlinear acoustic waves for cases with Kn = 10−1 and 10−2, equations The pressure is then calculated by using the ideal gas equation based on kinetic theory have been used. In the following, of state. The collision frequency is calculated at each time step we introduce the Bhatnagar-Gross-Krook (BGK) equation, a at every location in the physical space as model Boltzmann equation which is characterized by the use p of a linearized model for the molecular collisions in Sec. IV A, ν = , μ (20) and the numerical scheme used to solve it in Sec. IV B. where the dynamic viscosity, μ, for the gas is calculated using A. Mathematical model the power law described in Sec. III A. The Boltzmann transport equation governs the evolution The BGK model for the collision operator also conserves of the density distribution function, f (x , c , t ), where c refers mass, momentum, and energy and preserves the Euler limit in to the molecular velocity space. In general, the density distri- the Chapman-Enskog expansion [29]. While the Prandtl num- bution function is a seven-dimensional quantity and depends ber for Argon is 2/3, the BGK model equation naturally yields on three dimensions in physical space, x = (x, y, z), three a Prandtl number of 1 in the Chapman-Enskog expansion. components of the molecular velocity space, c = (cx, cy, cz ), For the sake of consistency, the Navier-Stokes simulations and time. The Boltzmann transport equation in such a case is have been run with unitary Prandtl number, despite being given by unphysical. Various numerical trials (not shown) reveal that / ∂ f varying the Prandtl number in the range from 2 3to1inthe + c · ∇ f = Q, Navier-Stokes simulations does not impact the conclusions of ∂ (15) t this manuscript nor yields significant quantitative changes in where Q is the collision operator. The Boltzmann collision the results. operator has the fundamental property of conserving mass, momentum and energy, that is B. Numerical scheme Q , θ = ,θ = , , | |2. ( f f ) (c)dc 0 (c) 1 c c (16) There are three broad categories of numerical schemes c ∈R3 to solve the Boltzmann equation: discrete velocity method For more details, the reader is referred to classic textbooks in (DVM) [35–38], Fourier spectral methods [39–42], and di- molecular gas dynamics [30]. rect simulation Monte Carlo (DSMC) [30,43–45]. We have For simplicity, we consider the BGK model [34]forthe utilized DVM [38] to solve the BGK equation, which involves Boltzmann transport equation restricted to one spatial dimen- discretization of the molecular velocity space. Consequently, sion, given by the governing equations and associated integrals, such as ∂ f ∂ f Eqs. (17) and (19), are solved for in a discrete sense. + c = ν( fγ − f ), (17) ∂t x ∂x We have used the second-order Runge-Kutta scheme to in- where fγ is the equilibrium distribution, given by the local tegrate the BGK model equation in time, utilizing a staggered Maxwellian distribution, flux conservative approach for the spatial discretization. The flux points are collocated at the cell faces and the solution ρ |c − u |2 M = − , points at the cell centers as shown in Fig. 4. The second-order (c) 3 exp (18) (2πRT ) 2 2RT upwind QUICK [46] scheme has been used for the evaluation

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Here, ξ j refers to the weights of the quadrature used for integration. This scheme was presented by Frezzotti [47] and has been evaluated computationally by Mieussens [48], Mieussens and Struchtrup [49], and Chigullapalli and Alex- eenko [50]. Boundary conditions As the problem considered here is one-dimensional in space, the only wall boundary conditions to be imposed are at the duct terminations, which we chose as adiabatic. This is implemented as specular boundary conditions [29]in the BGK solver, assuming total reﬂection with no diffusion effects from the wall and is modeled such that the values of f incident on the wall boundary are reﬂected back,

f [c j + (v wall · n )n ] = f [c i − (v wall · n )n ], (24) where the subscripts i and j represent the incident and re- flected quantities and v wall and n represent the velocity of the wall and wall normal, respectively. FIG. 5. Illustration of three grid sizes for the discretization of the molecular velocity space in cx and cr . V. RESULTS AND DISCUSSION of fluxes at the flux faces and can be formulated as As the piston-tube setup (Fig. 2) is excited at the first resonant frequency of the system, the acoustic amplitude of cx 6 3 1 the planar wave inside the tube increases until an equilibrium If cx|i+1/2 > 0:Fi+1/2 = fi + fi+1 − fi−1 , x 8 8 8 is reached between the large-scale energy injection into the system and the energy dissipation due to molecular collisions cx 6 3 1 or cx|i+1/2 0:Fi+1/2 = fi+1 + fi − fi+2 . (viscosity). Figure 6 shows the time series of (a) pressure, x 8 8 8 p, and (b) velocity, u, fluctuations at x = L and x = L/2, (21) = . = −5 respectively, for the case of Reup 0 2 and Kn 10 (see Table I) as obtained from the Navier-Stokes solver with Axial symmetry of the setup is used to reduce the molec- adaptive mesh refinement (NS-AMR). The insets (i) and (ii) ular velocity components to two—axial and radial. For ef- show the different stages of wave steepening before a limit ficiency and speed of calculations, we have used Gauss- cycle, or equilibrium limit, is reached. The insets (i) highlight Hermite and Gauss-Laguerre polynomials to obtain the dis- the harmonic regime in which the amplitude of the acoustic cretization points in the molecular velocity space in the axial wave is small relative to the base pressure and the linearized and radial directions, respectively. Different grid sizes for governing equations are sufficient. The insets (ii) show the integration in the molecular velocity space are shown in Fig. 5, nonlinear regime during which energy begins to cascade from where c and c refer to the molecular velocities in axial x r the fundamental resonant harmonic to higher harmonics in the and radial directions, respectively. The domain for molecular frequency domain. In the physical space, this results in wave velocity space used in this work is composed of 32 × 32 grid steepening. Continued injection of energy due to resonance points. results in saturation of the acoustic amplitude, characterized The equilibrium distribution function was discretized as by a balance between the energy injected into the system and β·p fγ = α1e , the thermoviscous dissipation—establishing an equilibrium energy cascade. Insets (iii) show this quasisteady regime, β = [−α ,α ,α ,α ], 2 3 4 5 hereafter referred to as the limit cycle. The latter cannot be 2 2 2 p ={[(cx − u) + (cy − v) + (cz − w) , ... considered an equilibrium state for the overall system given the steady background heating caused by the thermoviscous ...(c − u), (c − v), (c − w)]}T , (22) x y z dissipation (discussed later). Such slow-varying base-state where the coefficients αi are found by solving Eq. (16)using quantities are obtained via sharp spectral temporal filtering of Newton’s iteration method. The discrete form of the system of the numerical simulation data. equations solved is shown in Eq. (23): A. Comparison of Navier-Stokes and BGK solutions fγ ξ = ρ; c fγ ξ = ρu; j x j In this section we present a comparison between the results j j from the Navier-Stokes and BGK equation solvers in a Knud- cy fγ ξ j = ρv; cz fγ ξ j = ρw; sen number range typically considered of overlap between j j the two models. To this end, we show the temporal spectra of pressure and velocity at the limit cycle for the cases with 2 2 2 2 2 2 3 = −1 −2 c + c + c fγ ξ j = ρ(u + v + w ) + ρT. (23) Kn 10 and 10 . Since the signals are periodic at limit x y z 2 j cycle, p and u can be expressed as the following Fourier series

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= = / = . = −5 FIG. 6. Time series for (a) pressure fluctuations p at x L and (b) velocity u at x L 2forReup 0 2andKn 10 as obtained from the NS-AMR solver. The figure insets—(i), (ii)and(iii)—show the different stages in steepening of the planar acoustic wave until a limit cycle is achieved. expansions: the BGK solver for Kn = 10−1 and Kn = 10−2 and varying Reynolds numbers Reu . The pressure spectra are contiguous, k=∞ k=∞ p ω ω while only odd numbered harmonics appear in the velocity u(t ) = u(k)ei kt , p (t ) = p(k)ei kt , (25) spectra since the velocity signal is symmetric in time. k =−∞ k =−∞ The difference in the solutions obtained from the two k = 0 k = 0 solvers decreases as the Reynolds number and Knudsen num- whereu(k) and p(k) are coefficients of the kth harmonic mode ber decrease. Overall, the resonance response predicted by the of the Fourier series expansion with frequency ωk. The Fourier NS-AMR solver (which should only be applied in the dense coefficients are extracted via a windowed Fourier transform regime) is systematically lower than the one predicted from − − over 20 acoustic periods based on time series at x = L for p the BGK solver for Kn = 10 1 and Kn = 10 2. and at x = L/2foru (Fig. 6). Results from the NS-AMR solver seemingly under-predict − Figure 7 shows the comparison of the temporal spectra of acoustic energy levels for Kn 10 2 with respect to the BGK p and u as obtained from the NS-AMR solver with those from counterpart, and are hence neglected for this range. On the

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Re =0.2 10−4 Re u 10−4 up Reu Reu =0.02 p 10−7 Reu =0.002 10−7

10−10 10−10 2 2 2 0 2 0 a −13 a −13 0 10 0 10 ρ ρ / −16 / 2 −16 10 2 10 p p

10−19 10−19

10−22 10−22 Kn=10−1 Kn=10−2 10−25 10−25 100 101 100 101 ωk/ωres ωk/ωres (a) (b)

10−4 Re 10−4 Re up up

10−7 10−7

10−10 10−10 2 0 2 0 10−13 10−13 /a /a 2 2 u u 10−16 10−16

10−19 10−19

10−22 10−22 Kn=10−1 Kn=10−2 10−25 10−25 100 101 100 101 ωk/ωres ωk/ωres (c) (d)

FIG. 7. Power spectral densities of pressure at x = L (a, b) and velocity at x = L/2 (c, d) at the limit cycle for Kn = 10−1 and Kn = 10−2 from NS-AMR (black) and BGK (gray) solvers. Only the odd harmonics have been plotted for u. Spurious high-frequency or wavenumber content due to round-off errors has been omitted.

/| | = other hand, the use of the BGK solver will be restricted pressure perturbation (p p max) at the limit cycle for Reup to cases with Kn 10−2 (see Table III). Such choice also 0.2 when the piston is at its mean position moving away from reduces the computational cost of spanning the full parameter the cavity [ωrest = (2n + 1)3π/2inEq.(2)]. For Knudsen space given in Table I with the BGK solver, which suffers numbers Kn > 10−2 the pressure perturbation is qualitatively from the stiff collision term for Kn < 10−2. similar to a half cosine, with negligible to no nonlinear distortion. As the Knudsen number decreases (flow becomes B. Wave steepening and resonant response denser), the limit cycle pressure perturbation profile steepens Figure 8 shows the instantaneous waveform shape as further and a resonating shock wave is observable in the tube. A similar observation can be made in Fig. 8(b), which the Knudsen number of the system is varied, keeping Reup constant. Figure 8(a) shows the spatial profile of normalized shows the spatial profile of normalized velocity (u/|u|max)at TABLE III. Tabulation of the list of cases solved for by each solver: Navier-Stokes equations with adaptive mesh refinement (NS-AMR), Boltzmann equation with BGK closure (BGK).

←− Dense Rareﬁed −→ Kn = 10−5 Kn = 10−4 Kn = 10−3 Kn = 10−2 Kn = 10−1 = . Reup 0 2 NS-AMR NS-AMR NS-AMR BGK BGK = . Reup 0 02 NS-AMR NS-AMR NS-AMR BGK BGK = . Reup 0 002 NS-AMR NS-AMR NS-AMR BGK BGK

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1.5 103 Re up 2 1.0 10 Re =16.218 Umax

101 ReU =2.889 0.5 max

100 ReU =0.333 max max | 0.0 max p U | Kn /

−1 Re 10 p −0.5 Kn = 10−1 Kn = 10−2 10−2 Kn = 10−3 Re = 0.2 −1.0 up Kn = 10−4 −3 10 Reup = 0.02 Kn = 10−5 Re = 0.002 up Rarefied −→ −1.5 10−4 0.0 0.2 0.4 0.6 0.8 1.0 10−6 10−5 10−4 10−3 10−2 10−1 100 x/L Kn (a) (a) 1.2 105 1.0 Kn 104 0.8

0.6 103 p u max | Re u

| 0.4 / 102 Reu u/ p max 0.2 U Re 101 0.0

100 −0.2 Rarefied −→ 0.0 0.2 0.4 0.6 0.8 1.0 10−1 x/L 10−6 10−5 10−4 10−3 10−2 10−1 100 (b) Kn (b) FIG. 8. Instantaneous profiles of (a) pressure fluctuation, p,and (b) velocity, u, normalized by their maximum values at the limit cycle FIG. 9. Variation of ReUmax against Kn and Reup : (a) depicts the for Reu = 0.2. The pressure field is plotted at the instance when p behavior of ReUmax versus Knudsen number for all the cases con- the piston is at its mean position moving towards left, whereas the sidered and (b) illustrates ReUmax scaled by Reup versus the Knudsen velocity is taken at the time instance when the piston is at its right number. The degree of wave amplification due to resonance observed extremum. is the highest for the cases with the lowest Reup .

based on the maximum velocity amplitude in the domain, the limit cycle for Re = 0.2 when the piston is at its right up U , occurring at approximately x = L/2 for all cases. Re extremum [ω t = (2n + 1)π in Eq. (2)]. The velocity profile max Umax res characterizes the relative importance of inertial to viscous for Kn = 10−1 is qualitatively similar (but not exactly equal) forces more accurately than Re : the system can, in fact, to a half sine, whereas for decreasing Knudsen numbers, the up achieve an increase of three orders of magnitude in Re limit cycle velocity profile steepens and a shock wave is Umax with a four-orders-of-magnitude decrease in the Knudsen observed. Energy injection from the piston at the fundamental number, for the same Re [Fig. 9(a)]. Consequently, the harmonic frequency sustains the shock wave at the limit cycle, up magnitude of maximum velocity amplitude in the tube for a which, for the specific piston-Reynolds number considered in fixed piston velocity amplitude also increases with decreasing this example, Re = 0.2, is only present for Kn < 10−4. up Knudsen number, signifying a stronger resonant response of To more appropriately quantify the limit cycle amplitude the system [cf. Fig. 9(b)]. The resonant energy injected into dependency on the Knudsen number of the gas (cf. Fig. 9), the system is balanced by thermoviscous dissipation, which the following Reynolds number should be considered: increases with Knudsen number, resulting in an increase of the smallest length scale carrying acoustic energy, analogous ρ U L to hydrodynamic turbulence [11]; in other words, as the wave Re = 0 max , (26) Umax μ steepening process establishes an energy cascade towards

023101-9 THIRANI, GUPTA, AND SCALO PHYSICAL REVIEW E 101, 023101 (2020) higher wavenumbers, increasing the Knudsen number halts 10−3 the cascade at larger length scales, thus preventing further amplification of the acoustic wave due to resonance and reducing Kn the inertial range of inter-scale energy transfer (discussed later). In Fig. 9(a) three sets of cases, grouped by the same 10−4 . , . , value of ReUmax (16 218 2 889 and 0.333) for different values 2 0 a of Reup and Kn are highlighted and will be used, hereafter, 0 ρ when needed, to rigorously isolate Knudsen number effects / 10−5 from Reynolds number effects, the latter being ReUmax and not rms Reup . p Kn=10−1 Kn=10−2 Kn=10−3 C. Fluctuation intensity profiles −6 10 Kn=10−4 Besides affecting the resonant response, which is quanti- Kn=10−5 fied by the ratio of the maximum velocity fluctuation amplitude in the domain to the piston velocity amplitude [Fig. 9(b)], 0.0 0.2 0.4 0.6 0.8 1.0 the Knudsen number also affects the spatial pattern of fluctu- x/L ation intensities at the limit cycle. Figure 10 shows the spatial (a) profiles of intensity of pressure and velocity fluctuations, defined respectively as 10−3 Kn − N1 −4 1 10 p (x) = p(x, t )2, rms N n n=0 10−5 (27) − 0 N1 −6 /a 10 1 2 urms(x) = u(x, tn ) , rms

N u n=0 10−7 = . for Reup 0 02, and where N is the number of samples in 10−8 time over an acoustic period at the limit cycle. Using the Fourier coefficients and Parseval’s identity, the sum of squares 10−9 of point-wise pressure and velocity fluctuation values in the above equation can be evaluated as 10−10 0.0 0.2 0.4 0.6 0.8 1.0 N−1 N−1 x/L 2 = π | |2, p (tn ) 2 p(k) (b) n=0 k=0 (28) N−1 N−1 FIG. 10. Spatial variation of (a) pressure fluctuation intensity, 2 2 u(tn ) = 2π |u(k)| . prms, and (b) velocity fluctuation intensity, urms, at the limit cycle = . n=0 k=0 versus x for Reup 0 02. For increasing Knudsen number, the prms profile becomes sharper at the center indicating decreased movement With increasing Knudsen number, the spatial profile of of the pressure node at the center. pressure fluctuation amplitude displays a progressively more defined pressure node at the center for increasing Kn, which is associated to a decrease in acoustic power emanating from (taken at x = L) and u2 (taken at x = L/2), respectively, at the piston and being transmitted through the tube. For very = . the limit cycle for varying Knudsen numbers for Reup 0 2. low Knudsen numbers, the intensity profile remains relatively We note that the range of frequencies spanned by both sets of flat due to the redistribution of pressure fluctuation inten- spectra increases with decreasing Knudsen number displaying sity by the resonating shock wave. With increasing Knudsen a −2 slope over more than a decade on a log-log scale at the number, the deviation of the velocity fluctuation intensity limit cycle. A similar behavior is observed in hydrodynamic profile from the fluctuation intensity at the piston decreases turbulence, where such range of scales is associated with indicating less fluctuation amplification and hence a weaker inviscid spectral energy cascade in equilibrium conditions. resonant response. Hereafter, this range of scales will be referred to as the In the following, we study the distribution of temporal inertial range. However, in nonlinear acoustics, such power Fourier coefficients with varying Knudsen number, to quantify law range of spectra exhibits thermoviscous dissipation as the degree of nonlinear wave steepening and the consequent well, unlike hydrodynamic turbulence, as shown by Gupta higher harmonic generation. and Scalo [11]. The highest energy-containing frequency in the spectra is associated with the smallest length scale in the D. Spectral analysis and scaling of spectral quantities flow, which corresponds to the thickness of the shock (when Figure 11 shows the variation of distribution of temporal present). As discussed previously, with increasing Knudsen Fourier coefficients of pressure and velocity fluctuations, p2 number the dissipation increases resulting in a decrease in the

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where the energy correction term f (p)isgivenby 10−2 −2 Kn = 10−1 −2 2γ Kn = 10 f (p) =− 10−6 Kn = 10−3 (γ − 1)(2γ − 1) Kn = 10−4 1/γ 2 −10 Kn = 10−5 γ p p (γ −1)p 10 × + − − + . 2 1 1 (30) ρ a2 ρ a2 2 2 2 0 0 0 ρ 0 0 2 0a0 a 10−14 0 ρ Kn Utilizing the second-order perturbation energy [Eq. (29)]

/ −18 2 10 we deﬁne a characteristic dimensionless perturbation p amplitude, Arms,as 10−22 Arms = E, (31) 10−26 where . is the spatial averaging operator, −30 10 1 L 100 101 102 .= (.)dx. (32) ωk/ωres L 0 (a) Since the generation and dissipation of energy are balanced at the limit cycle, the cycle-averaged value of the perturbation 10−2 −2 energy [Eq. (29)] reaches a maximum in time. Balancing the characteristic rate of energy transfer from large scales to small −6 10 scales, with the rate of viscous energy dissipation, yields the following order-of-magnitude equivalence [11]: 10−10 2 Arms Arms 2 0 −14 ∼ δ , (33) 10 η η2 /a

2 Kn u 10−18 where η ∼ δ/Arms is the acoustic Kolmogorov length scale, i.e., the shortest length scale, analogous to the Kolmogorov 10−22 length scale in turbulence, carrying acoustic energy; δ is the thermoviscous diffusivity defined as 10−26 μ 4 γ − 1 δ = + , (34) 10−30 ρ0a0L 3 Pr 100 101 102 ω /ω where Pr is the Prandtl number. Analogously to the shortest k res τ (b) length scale, the smallest timescale, η, can be evaluated as η τ = . FIG. 11. Comparison of the temporal spectra at the limit cycle η (35) a0 for Re = 0.2 (a) for pressure at x = L and (b) velocity at x = up L/2, for which only the odd harmonics are shown. Spurious high- Utilizing Eq. (29), the spectral energy Ek can be defined frequency or wavenumber content due to round-off errors has been as [11] omitted. 2 2 |u(k)| |p(k)| f (p ) Ek = + + p(−k) . (36) 2a2 ρ 2 2 p 0 2 0a0 k highest acoustic frequency generated, shortening the inertial Figure 12 (top row) shows the distribution of spectral range; this is linked to an effective reduction of the Re . Umax energy Ek with the frequencies scaled by the fundamental Consequently, as shown in Fig. 8 the waveform at limit cycle frequency of the system. In each subfigure we compare cases exhibits energy only in larger length scales (it is less steep- with same ReUmax , which actually quantifies the limit cycle ened) for higher Knudsen numbers, if Reup is kept constant. energy in the system. Figure 12 (bottom row) shows the scaled As shown by Gupta and Scalo [11], the smallest length −2 spectral energy EkArms versus frequency scaled with the small- scale generated due to nonlinear wave steepening can be τ est timescale η for the three sets of cases grouped by ReUmax . estimated by closing the spectral energy budgets of wave per- We note that scaling the frequency with smallest timescale turbation. The perturbation energy for second-order nonlinear τη reveals an almost exclusively sub-Kolmogorov acoustic acoustic waves yielding the Lyapunov function of the system < energy range for ReUmax 10. This provides a quantitative is given by [11,51] estimate of when to expect the formation of an inertial range > i.e., ReUmax 100, as achieved in the cases with lower Kn in Fig. 11, characterized by the universal −2 slope of the u2 p2 E(p, u) = + + f (p), (29) inertial range. The latter is never actually observable in the 2a2 ρ 2 2 0 2 0a0 cases shown in Fig. 12.

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100 100 100 −2 10−5 10−5 10−5

10−10 10−10 10−10

k 10−15 10−15 10−15 E Re =16.218 Re =2.889 Re =0.333 Umax Umax Umax 10−20 10−20 10−20 Kn=10−3 Kn = 10−2 Kn = 10−1 10−25 Kn=10−4 10−25 Kn = 10−3 10−25 Kn = 10−2 Kn=10−5 Kn = 10−4 Kn = 10−3 10−30 10−30 10−30 100 101 100 101 100 101 ωk/ωres ωk/ωres ωk/ωres

1012 1012 1012 −2 107 107 107

102 102 102 2 − rms 10−3 10−3 10−3 A k E 10−8 10−8 10−8

10−13 10−13 10−13 Re =16.218 Re =2.889 Re =0.333 Umax Umax Umax 10−18 10−18 10−18 10−1 100 101 102 10−1 100 101 102 10−1 100 101 102 ωkτη/2π ωkτη/2π ωkτη/2π

= . = / FIG. 12. Comparison of the temporal spectra of Ek at the limit cycle for ReUmax 16 218 (left); 2.889 (mid); 0.333 (right) at x L 2. Spurious high-frequency or wavenumber content due to round-off errors has been omitted.

The spectra are collapsed along the frequency axis via the steepness of the waveform as shown in Fig. 8. The range the normalization ωkτη, but not in amplitude by Arms.This of wavenumbers spanned by both sets of spectra increases indicates that the (dimensionless) bandwidth of the spectral with decreasing Knudsen number displaying a −2 slope energy cascade increases with ReUmax . over more than a decade on a log-log scale at the limit To perform a wavenumber analysis of the spatial profiles cycle. of p and u, a complex Fourier series expansion can not be Utilizing Eq. (29), the spectral energy in wavenumber used. Due to the computational setup considered, cosine and space, Esp,k, can be written as [11] sine series expansions are used for the pressure and velocity, respectively. The corresponding spatial Fourier coefficients, P and U,ofp and u, respectively, can then be evaluated as 2 2 |U (k)| |P(k)| f (p ) Esp,k = + + P(−k) . N−1 2 2 2 π 1 2a0 2 ρ0a p P(k) = p(x ) cos n + k , 0 k n N 2 n=0 (39) k = 0, ....., N − 1, (37) N− Figure 14(a) shows the distribution of cycle-averaged E , 1 π 1 sp k U(k) = p(x )sin n + (k + 1) , with the wavenumber scaled by the length of the tube, for n N 2 n=0 the three sets of cases with same ReUmax at the limit cycle. k = 0, ....., N − 1, (38) Figure 14(b) shows the distribution of cycle-averaged scaled −2 η spectral energy, Esp,kArms, versus scaled wavenumber, k ,for where N is equal to the number of points in the spatial grid. the three sets of cases with same ReUmax at the limit cycle. As Figures 13(a) and 13(b) show the variation of distribution observed in the temporal spectra (Fig. 12), the highest scaled of cycle-averaged spatial Fourier coefficients of pressure and wavenumber, kη, is much larger than unity, confirming that the velocity fluctuations, P2 and U2, respectively, at the limit acoustic activity is in the sub-Kolmogorov range in space and = . = cycle for varying Knudsen numbers for Reup 0 2. Similar that ReUmax 16 is not high enough to yield a proper inertial to the trend observed for temporal spectra (Fig. 11), the range range (with −2 slope). of wavenumbers spanned by both sets of spectra increases To quantify the rate of spectral energy cascade, we define with decreasing Knudsen number indicating an increase in the spectral energy flux, k, from wavenumbers |k | k to

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10−2 10−2 1.50 −2 −2 10−5 10−5 1.25 Kn −8 −8 10 Kn 10 Kn 1.00 2 )

2 0 −11 −11 3 10 2 0 10 a − rms 0 0.75 /a ρ A 2 ( −1 −14 Kn = 10 −14 k /

10 U 10 2 −2 Π 0.50 P Kn = 10 10−17 Kn = 10−3 10−17 0.25 Kn = 10−4 10−20 10−20 −5 Kn = 10 0.00 10−23 10−23 100 101 102 100 101 102 100 101 102 kL kL kL (a) (b) (c)

FIG. 13. Comparison of the cycle-averaged spatial spectra of (a) p (x)and(b)u(x), and cycle-averaged (c) spectral ﬂux, k , scaled by Arms, = . at the limit cycle for Reup 0 2. Spurious high-frequency or wavenumber content due to round-off errors has been omitted. The ReUmax values = −1, −2, −3, −4 −5 . , . , . , . , = . for Kn 10 10 10 10 , and 10 are 0 396 3 153 16 252 54 518 and 174.571, respectively, at the limit cycle for Reup 0 2.

(a) 10−3 −2 10−3 10−3

10−6 10−6 10−6

10−9 10−9 10−9 ,k

sp 10−12 10−12 10−12 E Re =16.218 Re =2.889 Re =0.333 Umax Umax Umax 10−15 10−15 10−15 Kn=10−3 Kn=10−2 Kn=10−1 −4 −3 −2 10−18 Kn=10 10−18 Kn=10 10−18 Kn=10 Kn=10−5 Kn=10−4 Kn=10−3 10−21 10−21 10−21 100 101 100 101 100 101 kL kL kL (b) 102 102 102

100 −2 100 100

10−2 10−2 10−2

2 −4 −4 −4

− rms 10 10 10 A

,k −6 −6 −6 sp 10 10 10 E 10−8 10−8 10−8

−10 −10 −10 10 Re =16.218 10 Re =2.889 10 Re =0.333 Umax Umax Umax 10−12 10−12 10−12 10−1 100 101 102 10−1 100 101 102 10−1 100 101 102 kη kη kη (c) 1.0 0.4 0.20 Re =16.218 Re =2.889 Re =0.333 Umax Umax Umax 0.8 0.3 0.15

0.6 3

− rms 0.2 0.10 A k Π 0.4

0.1 0.05 0.2

0.0 0.0 0.00 10−1 100 101 102 10−1 100 101 102 10−1 100 101 102 kη kη kη

FIG. 14. Comparison of the cycle-averaged spatial spectra of (a, b) Esp,k and cycle-averaged (c) spectral ﬂux, k, at the limit cycle for = . ReUmax 16 218 (left); 2.889 (mid); 0.333 (right). Spurious high-frequency or wavenumber content due to round-off errors has been omitted.

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|k | > k as [11] 1400 ⎡ Kn = 10−3 −4 Kn = 10 ∂ ∂ P−k 1 (u g) P−k u p ) 1200 −5 = ⎣ + 5 Kn = 10 k ρ 2 ∂ ρ 2 ρ 3 ∂ − | | 0a0 a0 x 0a0 0a0 x k k k k 10 ⎤ 2 1000 Kn 2 2 2 0 − ⎥ a Kn + 1 U k ∂ u − p , 800 ⎦ ref ∂x 2 2 (40) ρ 2 ρ 2 a0 a0 ( 0a0 ) k res 600 ω ( where the function g is defined as [11] π/ 2 400 1/γ γ γ p p ac = + − − . ˙ g(p ) 1 1 (41) W γ − ρ 2 ρ 2 200 1 0a0 0a0 Figure 13(c) shows the distribution of cycle-averaged spec- 0 −3 0.0 0.2 0.4 0.6 0.8 1.0 tral energy flux, k, scaled by Arms, versus the wavenumber scaled by the length of the tube, at limit cycle for cases x/L = . with Reup 0 2. Figure 14(c) shows the distribution of cycle- − FIG. 15. Spatial variation of scaled mean acoustic power W˙ for averaged spectral energy flux, , scaled by A 3 ,versus ac k rms the cases with Re = 16.218. The decreasing trend of the acoustic scaled wavenumber, kη, for the three sets of cases with same Umax power with x indicates that the acoustic power is directed towards the Re at the limit cycle. The high negative gradient of the Umax right of Fig. 2 with influx of acoustic energy at the left end from the distribution implies that all of the acoustic energy in the piston. system is exhibited within the dissipation range of the spectra, and the absence of an inertial range for these sets of cases. The −3 equation for the slowly increasing base pressure, p0(τ ), we trend observed for the maximum value of kArms at lower wavenumbers further confirms that a proper inertial range apply a sharp spectral filter in time to Eq. (43). For a generic − > variable ϕ, the sharp spectral filter is defined as (with 2 slope) will be observed only for ReUmax 100 [see Fig. 13(c)]. ϕ = F(ϕ; ωcut ),ωcut <ωres, (44) The acoustic energy dissipated by thermoviscous dissipation is converted into base internal energy and leads to where F is the sharp spectral filter operator with cutoff frequency ωcut and ϕ can be decomposed into its mean an increase in the base-state pressure and temperature as mentioned before in Sec. II. In the subsection below, we and perturbed values as ϕ = ϕ + ϕ . Moreover, by definition F ϕ,ω = ϕ = quantify the Knudsen number effects on this process. ( cut ) 0. Substituting the decomposition [Eq. (3)] for p, u, and T , E. Base heating rates and thermal energy budgets and applying the sharp spectral filter to Eq. (43), we get The base pressure, p0, and temperature, T0, of the system ∂ p ∂(pu) 4 ∂u 2 ∂ p 0 + γ = (γ − ) μ + (γ − )u . increase slowly in time due to the conversion of acoustic ∂τ ∂ 1 ∂ 1 ∂ energy to heat by thermoviscous dissipation. To quantify the x 3 x x Knudsen number dependence of such heating, we evaluate (45) the rate of change in internal energy with the slow timescale, The thermal conduction term in Eq. (43) drops out upon τ, introduced in Sec. II. The governing equation for spatio- filtering since all base-state quantities are considered spatially temporal evolution of internal energy for a compressible fluid uniform (see Sec. II). Defining the mean acoustic power, W˙ac, in one dimension can be derived from Eqs. (8) and (9)as dissipation rate, , and mean mechanical work, ,as 2 2 ∂(ρe) ∂(ρue) μCp ∂ T 4 ∂u ∂u + = + μ − p , (42) 4 ∂u 2 ∂ p ∂t ∂x Pr ∂x2 3 ∂x ∂x W˙ = (pu),= μ ,= u , (46) ac 3 ∂x ∂x where e is the internal energy and Cp is the specific heat Eq. (45) can be compactly written as capacity at constant pressure. Using Cv = R/(γ − 1) and the ideal gas equation of state, Eq. (42) can be modified to ∂ p0 ∂ + γ W˙ac = (γ − 1) + (γ − 1). (47) 2 ∂τ ∂ ∂ p ∂(pu) ∂ μC ∂T 4 ∂u x + = (γ − 1) p + (γ − 1) μ ∂t ∂x ∂x Pr ∂x 3 ∂x Equation (47) yields the rate of change of base pressure, p , with the slow time, τ, in terms of cycle-averaged acous- ∂ pu ∂ p 0 + (1 − γ ) + (γ − 1)u . (43) tic power, viscous dissipation rate, and the cycle-averaged ∂x ∂x mechanical work. Figure 15 shows the spatial variation of Equation (43) governs the rate of change of pressure in the mean acoustic power, W˙ac, or acoustic energy flux. The time due to thermal conduction, viscous dissipation, and negative gradient of acoustic energy flux, see Fig. 16(a),is acoustic nonlinearities. As mentioned before [see Eq. (3)], p, due to the energy injection by the piston at the left end of T and u can be decomposed into rapidly fluctuating acoustic the tube. The acoustic energy flux is higher away from the quantities and slowly evolving base quantities. To obtain an piston for denser regimes indicating that the acoustic energy is

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3 3 3 ×10 1.2 ×10 ×10 ) 5 ) −3 ) 1 5 5 − 6 Kn=10 − −

10 −4 2 10

Kn=10 1.0 10 2 5 −5 2

Kn Kn=10 2 0 Kn Kn 2 0 2 0 a 0.8 0 a 4 a ref ref ref ρ ρ ρ

res Kn res 3 0.6 res ω ( ω ω ( ( −1 π/ Kn π/ 2 0.4 π/ )2 ac

˙ 1 1)Γ 2 W 1)Φ 2

( 0.2

− −2 ∂ − ∂x γ γ 0 γ ( (

− 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x/L x/L x/L (a)(b)(c)

FIG. 16. Spatial variation of (a) scaled mean acoustic power derivative ∂W˙ac/∂x, (b) dissipation , and (c) acoustic mechanical work = . versus x for the set of cases with ReUmax 16 218. transferred farther in denser gases, i.e., the acoustic near-field propagation. The setup was driven to a limit cycle, character- extends further in denser media. Figures 16(b) and 16(c) show ized by a balance between the resonant energy production and the scaled dissipation and acoustic mechanical work terms in thermoviscous dissipation. The resonant amplification and Eq. (47). Since the velocity gradients are maximum at the hard limit cycle were simulated using the numerical discretization end walls, the viscous dissipation term, , is maximum at of the Navier-Stokes equations for low Knudsen number the ends of the tube and increases with the Knudsen number cases and the Boltzman equation with BGK closure for high as Kn2. Moreover, the magnitude of the acoustic mechanical Knudsen number cases. work is maximum near the center of the tube since the pres- The dissipation increases with the degree of rarefaction of sure gradient and velocity amplitudes peak near the center. the gas in the tube. Due to increased dissipation, the resonant Assuming that the base pressure p0 is only a function of response and the waveform steepening at the limit cycle is time, taking the spatial average of Eq. (47) yields attenuated. Keeping the Reynolds number associated to the piston constant, we observed that with increasing the Knudsen dp0 = + γ − + γ − , W˙in ( 1) L ( 1) L (48) number, the degree of resonance decreases, i.e., the ratio of dτ acoustic velocity amplitude to the piston velocity amplitude where decreases. Moreover, the pressure intensity profiles show that 1 L ∂ the pressure node, or the zero-crossing of the acoustic pres- W˙in =− γ W˙ac dx =−γW˙p, L 0 ∂x sure tends to become quasi-stationary for very high Knudsen L L numbers thus suggesting that in highly rarefied gas regimes, 1 1 the system behaves as a quasilinear damped oscillator even at L = dx,L = dx. (49) L 0 L 0 limit cycle. Due to an increase in dissipation with increasing Knudsen Here, W˙ refers to the acoustic power injected into the in number, the smallest length scale (diffusion length scale) system by the piston and is proportional to the mechanical which exhibits acoustic energy also increases. Consequently, work done by the piston, W˙ . The acoustic energy injected p the acoustic wave at the limit cycle becomes less steepened into the system propagates towards the fixed end of the tube with increasing Knudsen number. Since the nonlinear acoustic with the gradient of this flux increasing with the degree of wave steepening entails acoustic energy cascade into higher nonlinearity in the system. The base heating effect is produced harmonics and hence spectral broadening, the range of the by the terms, W˙ and in Eq. (48), which are positive in L acoustic energy spectrum decreases with increasing Knudsen and increase with Knudsen number (see Fig. 16), suggesting number. We studied both the temporal and spatial spectra that the rate of increase in the base pressure of the system is of the acoustic energy with varying Knudsen numbers. Fur- proportional to the Knudsen number of the system. thermore, we showed that the largest harmonic or wavenumber can be estimated using the smallest length scale due to VI. CONCLUSION nonlinear acoustic steepening, which can be approximated We studied the effects of gas rarefaction on finite amplitude by balancing the nonlinear wave propagation and dissipation planar acoustic waves generated in a resonator by varying the terms. Scaling the spectra with the smallest length scale or Knudsen number over four orders of magnitude. The setup timescale reveals almost exclusively sub-Kolmogorov acous- studied consisted of a piston-tube assembly with a piston tic activity for Reynolds numbers based on the maximum oscillating at fundamental resonant frequency of the tube, velocity amplitude, ReUmax , less than 100 and provides a quan- closed at the other end. The tube was modeled as a one- titative metric of when to expect the formation of an inertial dimensional domain neglecting the effects of shear on wave subrange.

023101-15 THIRANI, GUPTA, AND SCALO PHYSICAL REVIEW E 101, 023101 (2020)

At limit cycle, the energy due to resonance sustains the ACKNOWLEDGMENTS nonlinear waves which dissipate the energy at length scales The authors acknowledge the support of the National depending on the Knudsen numbers. We showed that the Science Foundation (NSF) Fluid Dynamics Program (Award dissipation of energy at the limit cycle increases the mean No. 1706474) and the Air Force Office of Scientific Re- internal energy of the system slowly. We quantified this mean search (AFOSR) 2018 Young Investigator Award (YIP) (No. heating of the system utilizing time filtered internal energy FA9550-18-1-0292). Computational resources are provided budget equation. The results from the budget equation also by the Rosen Center for Advanced Computing (RCAC) at reveal the scaling of dissipation with Knudsen number as Kn2 Purdue University. for a fixed ReUmax .

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