Spectral energy cascade and decay in nonlinear acoustic waves

We present a numerical and theoretical investigation of nonlinear spectral energy cascade of decaying finite-amplitude planar acoustic waves in a single-component ideal gas at standard temperature and pressure (STP). We analyze various one-dimensional canonical flow configurations: a propagating traveling wave (TW), a standing wave (SW), and randomly initialized Acoustic Wave Turbulence (AWT). We use shock-resolved mesh-adaptive direct numerical simulations (DNS) of the fully compressible one-dimensional Navier-Stokes equations to simulate the spectral energy cascade in nonlinear acoustic waves. We also derive a new set of nonlinear acoustics equations truncated to second order and the corresponding perturbation energy corollary yielding the expression for a new perturbation energy norm $E^{(2)}$. Its spatial average, satisfies the definition of a Lyapunov function, correctly capturing the inviscid (or lossless) broadening of spectral energy in the initial stages of evolution -- analogous to the evolution of kinetic energy during the hydrodynamic break down of three-dimensional coherent vorticity -- resulting in the formation of smaller scales. Upon saturation of the spectral energy cascade i.e. fully broadened energy spectrum, the onset of viscous losses causes a monotonic decay of in time

Linear Stability Analysis of Compressible Channel Flow with Porous Walls

We have investigated the effects of permeable walls, modeled by linear acoustic impedance with zero reactance, on compressible channel flow via linear stability analysis (LSA). Base flow profiles are taken from impermeable isothermal-wall laminar and turbulent channel flow simulations at bulk Reynolds number, $Re_b$= 6900 and Mach numbers, $M_b$ = 0.2, 0.5, 0.85. For a sufficiently high value of permeability, Two dominant modes are made unstable: a bulk pressure mode, causing symmetric expulsion and suction of mass from the porous walls (Mode 0); a standing-wave-like mode, with a pressure node at the centerline (Mode I). In the case of turbulent mean flow profiles, both modes generate additional Reynolds shear stresses augmenting the (base) turbulent ones, but concentrated in the viscous sublayer region; the trajectories of the two modes in the complex phase velocity space follow each other closely for values of wall permeability spanning two orders of magnitude, suggesting their coexistence. The transition from subcritical to supercritical permeability does not alter the structure of the two modes for the range of wavenumbers investigated, suggesting that wall permeability simply accentuates pre-existing otherwise stable modes. Results from the present investigation will inform the design of new compressible turbulent boundary layer control strategies via assigned wall-impedance.Comment: 18 pages, 8 figures, Invited contribution to the Whither Turbulence and Big Data in the 21st Century Springer Volum

Universal Scaling of Acoustic and Thermoacoustic Waves in Compressible Fluids

We have derived the set of reference scaling parameters yielding collapse of isentropic acoustic and thermoacoustic (or heat-release-induced) waves across different pure compressible fluids with an assigned equation of state. The resulting reference pressure and velocity are consistent with classic acoustic scaling. The reference temperature and heat release rate need to be expressed in terms of the isobaric thermal expansion coefficient $\alpha^*_{p_0}$ to ensure collapse of all thermo-fluid-dynamic fluctuations across all fluids. The proposed scaling is extended to non-isentropic waves and verified against data from highly-resolved one-dimensional Navier-Stokes simulations. Conditions tested include freely propagating isentropic acoustic waves and thermoacoustic compression waves up to shock strength of 4.27, for six different supercritical fluids in states ranging from compressible liquid to near-ideal gas, and spanning seven orders of magnitude of imposed heat release rate

Acoustic Impedance Calculation via Numerical Solution of the Inverse Helmholtz Problem

Assigning homogeneous boundary conditions, such as acoustic impedance, to the thermoviscous wave equations (TWE) derived by transforming the linearized Navier-Stokes equations (LNSE) to the frequency domain yields a so-called Helmholtz solver, whose output is a discrete set of complex eigenfunction and eigenvalue pairs. The proposed method -- the inverse Helmholtz solver (iHS) -- reverses such procedure by returning the value of acoustic impedance at one or more unknown impedance boundaries (IBs) of a given domain via spatial integration of the TWE for a given real-valued frequency with assigned conditions on other boundaries. The iHS procedure is applied to a second-order spatial discretization of the TWEs derived on an unstructured grid with staggered grid arrangement. The momentum equation only is extended to the center of each IB face where pressure and velocity components are co-located and treated as unknowns. One closure condition considered for the iHS is the assignment of the surface gradient of pressure phase over the IBs, corresponding to assigning the shape of the acoustic waveform at the IB. The iHS procedure is carried out independently for each frequency in order to return the complete broadband complex impedance distribution at the IBs in any desired frequency range. The iHS approach is first validated against Rott's theory for both inviscid and viscous, rectangular and circular ducts. The impedance of a geometrically complex toy cavity is then reconstructed and verified against companion full compressible unstructured Navier-Stokes simulations resolving the cavity geometry and one-dimensional impedance test tube calculations based on time-domain impedance boundary conditions (TDIBC). The iHS methodology is also shown to capture thermoacoustic effects, with reconstructed impedance values quantitatively in agreement with thermoacoustic growth rates.Comment: As submitted to the Journal of Sound and Vibration (Elsevier) -- Updated 01/06/18. arXiv admin note: substantial text overlap with arXiv:1708.0065

High-fidelity simulation of a standing-wave thermoacoustic-piezoelectric engine

We have carried out wall-resolved unstructured fully-compressible Navier--Stokes simulations of a complete standing-wave thermoacoustic piezoelectric (TAP) engine model inspired by the experimental work of Smoker et al. (2012). The model is axisymmetric and comprises a 51 cm long resonator divided into two sections: a small diameter section enclosing a thermoacoustic stack, and a larger diameter section capped by a piezoelectric diaphragm tuned to the thermoacoustically amplified mode (388 Hz). The diaphragm is modelled with multi-oscillator broadband time-domain impedance boundary conditions (TDIBCs), providing higher fidelity over single-oscillator approximations. Simulations are first carried out to the limit cycle without energy extraction. The observed growth rates are shown to be grid-convergent and are verified against a numerical dynamical model based on Rott's theory. The latter is based on a staggered grid approach and allows jump conditions in the derivatives of pressure and velocity in sections of abrupt area change and the inclusion of linearized minor losses. The stack geometry maximizing the growth rate is also found. At the limit cycle, thermoacoustic heat leakage and frequency shifts are observed, consistent with experiments. Upon activation of the piezoelectric diaphragm, steady acoustic energy extraction and a reduced pressure amplitude limit cycle are obtained. A heuristic closure of the limit cycle acoustic energy budget is presented, supported by the linear dynamical model and the nonlinear simulations. The developed high-fidelity simulation framework provides accurate predictions of thermal-to-acoustic and acoustic-to-mechanical energy conversion (via TDIBCs), enabling a new paradigm for the design and optimization of advanced thermoacoustic engines

Bulk viscosity model for near-equilibrium acoustic wave attenuation

Acoustic wave attenuation due to vibrational and rotational molecular relaxation, under simplifying assumptions of near-thermodynamic equilibrium and absence of molecular dissociations, can be accounted for by specifying a bulk viscosity coefficient $\mu_B$. In this paper, we propose a simple frequency-dependent bulk viscosity model which, under such assumptions, accurately captures wave attenuation rates from infrasonic to ultrasonic frequencies in Navier--Stokes and lattice Boltzmann simulations. The proposed model can be extended to any gas mixture for which molecular relaxation timescales and attenuation measurements are available. The performance of the model is assessed for air by varying the base temperature, pressure, relative humidity $h_r$, and acoustic frequency. Since the vibrational relaxation timescales of oxygen and nitrogen are a function of humidity, for certain frequencies an intermediate value of $h_r$ can be found which maximizes $\mu_B$. The contribution to bulk viscosity due to rotational relaxation is verified to be a function of temperature, confirming recent findings in the literature. While $\mu_B$ decreases with higher frequencies, its effects on wave attenuation become more significant, as shown via a dimensionless analysis. The proposed bulk viscosity model is designed for frequency-domain linear acoustic formulations but is also extensible to time-domain simulations of narrow-band frequency content flows.Comment: Submitted manuscrip

Coherent-vorticity Preserving Large-Eddy Simulation of trefoil knotted vortices

We have performed Coherent-vorticity Preserving Large-Eddy simulations of a trefoil knot-shaped vortex, inspired by the experiments of Kleckner and Irvine. The flow parameter space is extended in the present study, including variations of the circulation Reynolds numbers in the range Re = 2000 - 200000, where Re = 20000 is the value used in the experiments. The vortex line corresponding to the trefoil knot is defined using a parametric equation and the Biot-Savart law is employed to initialize the velocity field. The CvP LES computation displays a good qualitative match with the experiment. In particular, the vortex entanglement process is accurately represented as well as the subsequent separation of the main vortex in two distinct structures - a small and a large vortex - with different self-advection speeds that have been quantified. The small vortex propagates faster than the large oscillatory vortex which carries an important amount of vorticity. The advection velocity of the vortex before bursting is found to be independent of the Reynolds number. The low Reynolds number computation leads to a decrease of the separated vortices velocity after bursting, compared to the higher Reynolds computations. The computation of energy spectra emphasizes intense energy transfers from large to small scales during the bursting process. The evolution of volume-averaged enstrophy shows that the bursting leads to the creation of small scales that are sustained a long time in the flow, when a sufficiently large Reynolds number is considered (Re>20000). The low Reynolds number case Re = 2000 hinders the generation of small scales during the bursting process and yields essentially laminar dynamics. The onset of background turbulence due to the entanglement process can be observed at Re = 200000Comment: 10 pages, 2018 AIAA Scitec

Pseudophase-Change Effects in Turbulent Channel Flow under Transcritical Temperature Conditions

We have performed direct numerical simulations (DNS) of compressible turbulent channel flow at supercritical pressure with top and bottom isothermal walls kept respectively at a supercritical (Ttop > Tpb) and subcritical temperature (Tbot < Tpb), where Tpb is the pseudoboiling temperature. The DNS are conducted using a high-order discretization of the fully compressible Navier-Stokes equations in conservative form closed with the Peng-Robinsion (PR) state equation. Bulk density is adjusted to obtain a bulk pressure of approximately pb = 1.1pcr where pcr is the critical pressure of the working fluid. Top-to-bottom temperature differences investigated are DT = 5 K, 10 K, and 20 K, where Ttop/bot = Tpb +- DT / 2; buoyancy effects are neglected. Varying DT modifies the average location of pseudophase change from ypb/h = -0.23 (DT = 5 K) to 0.89 (DT = 20 K), where h is the channel half-height and y = 0 the centerline position. Real-fluid effects cause visible deviations from classical scaling laws in the mean velocity profile. Enstrophy generation due stretching and tilting decreases with DT. The proximity to the pseudotransitioning layer inhibits the intensity of the velocity fluctuations, while enhancing the density and temperature fluctuations. Conditional probability analysis reveals that the sheet of fluid undergoing pseudophase change is characterized by a dramatic reduction in the kurtosis of density fluctuations and becomes thinner as DT is increased. Instantaneous visualizations show dense fluid ejections from the pseudoliquid viscous sublayer, some reaching the channel core, causing positive values of density skewness in the respective buffer-layer region (vice versa for the top wall).Comment: 39 pages, 32 figures, 6 table

Linear and Nonlinear Modeling of a Traveling-Wave Thermoacoustic Heat Engine

We have carried out three-dimensional Navier-Stokes simulations, from quiescent conditions to the limit cycle, of a traveling-wave thermoacoustic heat engine (TAE) composed of a long variable-area resonator shrouding a smaller annular tube, which encloses the hot (HHX) and ambient (AHX) heat-exchangers, and the regenerator (REG). Simulations are wall-resolved, with no-slip and adiabatic conditions enforced at all boundaries, while the heat transfer and drag due to the REG and HXs are modeled. HHX temperatures have been investigated in the range 440K - 500K with AHX temperature fixed at 300K. The initial exponential growth of acoustic energy is due to a network of traveling waves amplified by looping around the REG/HX unit in the direction of the imposed temperature gradient. A simple analytical model demonstrates that such thermoacoustic instability is a Lagrangian thermodynamic process resembling a Stirling cycle. A system-wide linear stability model based on Rott's theory is able to accurately predict the frequency and the growth rate, exhibiting properties consistent with a supercritical Hopf bifurcation. The limit cycle is governed by acoustic streaming -- a rectified flow resulting from high-amplitude acoustics. Its key features are explained with an axially symmetric incompressible model driven by the wave-induced stresses extracted from the compressible calculations. These features include Gedeon streaming and mean recirculations due to flow separation. The first drives the mean advection of hot fluid from the HHX to a secondary heat exchanger (AHX2), in the thermal buffer tube (TBT), necessary to achieve saturation of the acoustic energy growth. The direct evaluation of the energy fluxes reveals that the efficiency of the device deteriorates with the drive ratio and that the acoustic power in the TBT is balanced primarily by the mean advection and thermoacoustic heat transport.Comment: 35 pages, 19 figure

A Coherent vorticity preserving eddy viscosity correction for Large-Eddy Simulation

This paper introduces a new approach to Large-Eddy Simulation (LES) where subgrid-scale (SGS) dissipation is applied proportionally to the degree of local spectral broadening, hence mitigated or deactivated in regions dominated by large-scale and/or laminar vortical motion. The proposed Coherent vorticity preserving (CvP) LES methodology is based on the evaluation of the ratio of the test-filtered to resolved (or grid-filtered) enstrophy $\sigma$. Values of $\sigma$ close to 1 indicate low sub-test-filter turbulent activity, justifying local deactivation of the SGS dissipation. The intensity of the SGS dissipation is progressively increased for $\sigma < 1$ which corresponds to a small-scale spectral broadening. The SGS dissipation is then fully activated in developed turbulence characterized by $\sigma \le \sigma_{eq}$, where the value $\sigma_{eq}$ is derived assuming a Kolmogorov spectrum. The proposed approach can be applied to any eddy-viscosity model, is algorithmically simple and computationally inexpensive. LES of Taylor-Green vortex breakdown demonstrates that the CvP methodology improves the performance of traditional, non-dynamic dissipative SGS models, capturing the peak of total turbulent kinetic energy dissipation during transition. Similar accuracy is obtained by adopting Germano's dynamic procedure albeit at more than twice the computational overhead. A CvP-LES of a pair of unstable periodic helical vortices is shown to predict accurately the experimentally observed growth rate using coarse resolutions. The ability of the CvP methodology to dynamically sort the coherent, large-scale motion from the smaller, broadband scales during transition is demonstrated via flow visualizations. LES of compressible channel are carried out and show a good match with a reference DNS