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1 Proceedings of Meetings on Acoustics Volume 9, 23 ICA 23 Montreal Montreal, Canada 2-7 June 23 Engineering Acoustics Session aea: Thermoacoustics I aea8. Computational fluid dynamics simulation of ayleigh streaming in a vibrating resonator Joris P. Oosterhuis*, Simon Bühler, Douglas Wilcox and Theo H. Van der Meer *Corresponding author's address: Thermal Engineering, University of Twente, P.O. Box 27, Enschede, 75AE, Overijssel, Netherlands, j.p.oosterhuis@utwente.nl ayleigh streaming is a time-averaged flow that can exist in the thermal buffer tubes of thermoacoustic prime movers and refrigerators and is driven by the viscous stresses close to the solid boundaries. This mean flow leads to mean convective heat transport, that can have large impact on the performance of thermoacoustic devices. ayleigh streaming in a standing wave resonator is simulated using a commercially available computational fluid dynamics (CFD) code and is compared to existing analytical models of Hamilton et al. (23). A test case is developed and a standing wave is generated by applying a harmonic volume force to the domain. Both the inner and outer streaming vortices are well described for a range of radii from /δν=3...2 and the magnitude of the streaming velocity matches analytical values. This paper shows the possibility of using available as-is CFD software for the simulation of streaming in a standing wave resonator. The presented results pave the way for the simulation of more complex geometries and studies to reduce the negative effects ayleigh streaming can have on thermo-acoustic prime mover and refrigerator efficiency. Published by the Acoustical Society of America through the American Institute of Physics 23 Acoustical Society of America [DOI:.2/.47995] eceived 4 Jan 23; published 2 Jun 23 Proceedings of Meetings on Acoustics, Vol. 9, 38 (23) Page
2 INTODUCTION In thermoacoustic devices the energy is carried by an acoustic wave. This acoustic wave can induce a time-averaged flow, or acoustic streaming, which can lead to undesired heat convection and loss of efficiency. Different classes of acoustic streaming can be distinguished based on the underlying physical mechanisms. ayleigh streaming, as first analytically described by Lord ayleigh in 884 [], is the phenomenon of a net mean flow in a standing wave resonator driven by the viscous stresses close to the solid boundary [2]. For a parallel plate resonator half channel width of y > 5.7 δ ν where δ ν is the viscous penetration depth, this results in two steady vortices: one vortex inside the viscous boundary layer and one vortex outside the boundary layer [3]. The streaming vortices outside the boundary layer are often referred to as outer streaming or ayleigh streaming while the streaming vortices inside the viscous boundary layer are referred to as inner streaming or Schlichting streaming, named after Hermann Schlichting who described the first analytical solution for the streaming vortices inside the boundary layer in 932 [2]. Acoustic streaming can have a large impact on the performance of thermoacoustic prime movers and refrigerators as the generated mean flow can lead to undesirable mean convective heat transport. In thermal buffer tubes ayleigh streaming leads to a transport of enthalpy along the walls of the tube in one direction and transport along the central part of the tube in counter direction [4]. Olsen & Swift first described the theory of a tapered tube geometry that eliminates ayleigh streaming [5]. As an extension of the classical streaming models [, 6, 7, 8], new models have been developed more recently that take into account the effect of temperature and have less restrictions on the geometry [3, 9,, ]. Hamilton et al. [3] have derived a model based on the leading-order approximation of the linearized Navier-Stokes equations where the assumption of y δ ν is dropped allowing for the calculation of ayleigh streaming profiles for very small plate spacings. Moreover, this model describes both the inner and outer vortices correctly, providing a complete model for a wide range of geometry sizes. The only restrictions are that the viscous penetration depth is very small compared to the acoustic wavelength (L ν/c ) and slow (linear) streaming is assumed. In a second paper, Hamilton et al. extended their model to include thermal effects [9]. This shows a difference of about 2% in the streaming velocities in the wide-channel limit (y /δ ν = 2) compared to the isothermal case. Although ayleigh streaming can be eliminated effectively in thermal buffer tubes and can be well described analytically for any arbitrary plate spacing, the ayleigh streaming profiles and corresponding elimination mechanisms remain unclear for more complex geometries [5, 9]. In order to predict acoustic ayleigh streaming in complex geometries as well as at higher pressure drive ratios with corresponding non-linear streaming phenomena, computational fluid dynamics (CFD) comes into focus as a solution. Before heading towards these complex situations, a numerical test case has been developed in order to characterize the feasibility and accuracy of simulating ayleigh streaming using CFD. In the present paper, ayleigh streaming in a standing wave resonator is simulated using commercially available CFD code. It is shown that available as-is CFD software is capable of describing ayleigh streaming accurately for a range of sizes in both a parallel plate and tubular geometry. Proceedings of Meetings on Acoustics, Vol. 9, 38 (23) Page 2
3 MODELING To eliminate other classes of acoustic streaming and calculate a net mean velocity field that is solely generated by ayleigh streaming, a pure standing wave needs to be generated inside the domain of interest. A resonator is used to allow a standing wave to be generated [2]. By axially vibrating this resonator, a standing wave will start to grow and after a certain time a steady periodic solution with a constant wave amplitude is obtained. From here on, the streaming profiles can be calculated by time-averaging the velocity and density fields. The geometry and simulation setup is described below. Geometry A resonator with length L = λ/2 is modeled as a 2D parallel plate configuration consistent to the geometry Hamilton et al. used in their derivation [3, 9]. Helium at T = 3 K and p = atm is used as a fluid and the vibration frequency is set to f = Hz. Hence λ = c /f =.9 m and the length of the resonator is L = λ/2 = 5.95 m. The viscous penetration depth δ ν for the current fluid and operating conditions is: δ ν = 2μ/ωρ = m () For meshing purposes an extended viscous boundary layer thickness is defined as the region with the highest gradients [2]: δ ν,ext = π δ ν =.8 3 m. As in current thermoacoustic devices a tubular geometry is more common than a parallel plate geometry, after validation with the analytical model of Hamilton et al. [9], the model is extended to describe a 2D axi-symmetric tube instead of parallel plates. All dimensions stay the same although the half plate spacing y is now replaced by the tube radius. The simulation results with this geometry are described in a separate paragraph. Numerical Setup For all simulations the finite-element based software package COMSOL Multiphysics v4.3 is used. Transient CFD simulations have been carried out using the non-isothermal flow module that models the unsteady Navier-Stokes equations [3]. As a time-dependent solver the fully coupled direct MUMPS solver is used with a constant time discretization of Δt = 5 5 s. This yields N t = 2 time-steps per wave period (with f = Hz). The spatial discretization is described in a separate paragraph. In all cases a laminar fluid model is used as the turbulent effects for these relative small radii and velocity amplitudes are expected to be negligible [5]. The total energy is solved including the viscous work term [3]. The domain is initialized with zero velocity, zero relative pressure and a temperature of T = 3 K. In all cases the ideal gas law is used as an equation of state. The resulting set of transport equations solved by COMSOL Multiphysics are the unsteady Navier-Stokes equations [3]. The governing equations in conservation form are: one continuity equation, momentum equation in three directions and one total energy equation respectively: ρ + (ρ u) t =, (2) ρ u t + ρ ( u ) u = ρ τ + F, (3) T ρc p t + ρc p u T = (k T) +Q +Q vh +W p, (4) Proceedings of Meetings on Acoustics, Vol. 9, 38 (23) Page 3
4 isothermal no-slip walls δ ν,ext L y symmetry L x FIGUE : Geometry of vibrating resonator with boundary conditions and important length scales indicated. Vibration is imposed along x-axis. Not to scale and mesh is only for illustration purposes. with τ the stress tensor, Q vh the contribution of viscous heating and W p the pressure work term [3]. C p is the specific heat and k is the thermal conductivity of, in this case, helium. These governing equations are converted to the corresponding weak formulation before solving. For further details about the weak formulation, the reader is referred to the COMSOL Multiphysics documentation [3]. Because of symmetry, only the upper half of the domain is simulated. Hence, a symmetry boundary condition is applied on the center axis at y =. On the three remaining sides, an isothermal no-slip wall boundary condition is applied resulting in a fully closed system identical to the conditions imposed by Hamilton et al. [3, 9]. In Figure the geometry with boundary conditions is shown. Excitation Method In order to shake the resonator with the desired frequency f, a time-dependent volume force in the axial direction, F x (t), is applied to the system by implementing a uniform time-dependent source term in the momentum equation (Eq. 3) and a corresponding source term in the energy equation (Eq. 4) [4]: F x (t) = ρ 2πf u cos(2πft), (5) Q(t) = ρ u 2πf u cos(2πft), (6) with u an initial velocity amplitude. The relation between u and the velocity amplitude in the steady periodic situation u is dependant on the resonator plate spacing y or the radius for a tubular configuration. The initial velocity amplitude u is chosen such that linear streaming can be expected. Streaming Velocity Calculation To visualize the ayleigh streaming velocities that are second order effects, all first order wave effects should be eliminated. Hence, the ayleigh streaming net mean velocity field is calculated from the numerical time-series results by time-averaging the results over an integer number of periods to eliminate the first order wave: = ρ u ρ. (7) In all the following results the axial streaming velocity is normalized by the peak streaming velocity u. This is the axial streaming velocity as obtained by ayleigh in the center of a channel Proceedings of Meetings on Acoustics, Vol. 9, 38 (23) Page 4
5 with the radius in the limit y /δ ν [3]: u = 3 u 2, (8) 6 c L and the radial streaming velocity will be normalized correspondingly using ˆ = u y. For small acoustic amplitudes, the streaming profiles can be described using linear equations. These so-called slow streaming models include all classical streaming models and neglect the effect of fluid inertia [5]. Menguy & Gilbert defined a non-linear eynolds number for streaming to distinguish between the linear and non-linear streaming regimes [5]: e nl = M2 Sh 2, (9) where M = u c is the acoustic Mach number and Sh = ν ω is the Shear number. For e nl the streaming is considered linear and fluid inertia may be neglected. As the goal of this paper is to validate the numerical results with an analytical model which is only valid for linear streaming, the non-linear eynolds number will be chosen such that e nl for all cases. Mesh Details In order to characterize the required spatial discretization, a mesh dependency study has been carried out. A base mesh of N x N y elements is used and subsequently refined in the extended boundary layer δ ν,ext near the solid walls. This is done by applying a geometric element distribution with a growth rate of r =.2 between two succeeding elements. The number of elements in the boundary layer refinement at the axial wall is indicated with N y,bl. At the left and right walls N x,bl elements are used to resolve the boundary layer. The smallest element size is determined by the number of elements applied in the (extended) boundary layer: Δy min = δ ν,ext /N y,bl. By comparing the absolute difference (L 2 -norm) between the numerical axial streaming velocity field and the analytical axial streaming velocity field u ex L 2 = N x N y N x N y i= 2 : (,i u ex 2,i) 2, () it is observed that a relative coarse mesh resolution of N x = 5, N x,bl = 5, N y = 5, N y,bl = 5, resulting in a total number of elements of N tot = 6, is sufficient to resolve the streaming vortices within an accuracy of L 2 < 5%. efining the mesh in the axial direction does not lead to a significant decrease in L 2. However, when looking at the overall axial streaming velocity field, it is observed that for a coarse (axial) discretization wiggles distort the smoothness of the solution. Therefore, it is decided to further refine the mesh in order to obtain smooth streaming vortices. This yields a mesh resolution of N x = 4, N x,bl = 2, N y = 2, N y,bl = corresponding to a total number of elements of N tot = 3,2. It must be emphasized that the coarse mesh does lead to correct streaming magnitudes and vortex locations and thus can be used for quick design optimization purposes. ESULTS For both the parallel plate geometry as well as the tubular geometry, four different domain sizes have been simulated in order to investigate the influence on the streaming patterns. The different Proceedings of Meetings on Acoustics, Vol. 9, 38 (23) Page 5
6 TABLE : Overview of all simulation cases with important parameters case y /δ ν [-] u [m/s] u [m/s] e nl [-] u [m/s] P P P P case /δ ν [-] u [m/s] u [m/s] e nl [-] u [m/s] T T T T half plate spacings y or radii are all chosen as a multiple of the viscous penetration depth: y /δ ν = 3,5,,2 and the meshes are adjusted correspondingly to capture all effects in the (extended) viscous boundary layer as described in the preceding paragraph. The excitation velocity u is chosen such that the streaming eynolds number is in the same order of magnitude for all different simulations and e nl. This is done to obtain slow (linear) streaming which allows for comparison with an analytical model. An overview of the different simulation cases with corresponding values for u, u, e nl and u is shown in Table. In the subsequent paragraphs the simulation results will be presented for each different geometry and both qualitatively and quantitatively compared to analytical results. In all cases the analytical streaming solution is calculated using the non-isothermal model of Hamilton et al. [9] by matching the first order velocity amplitude. For more details about the streaming model of Hamilton et al., the reader is referred to the corresponding papers [3, 9]. Parallel Plate Geometry Figure 2 shows the simulation results for the various parallel plate simulation cases P. P.4 compared to the analytical solution calculated with the non-isothermal streaming model of Hamilton et al. [9]. It is clear that for cases with y /δ ν < 5.7 (P. P.2), only the inner streaming vortices appear while for larger radii (P.3 P.4) both the inner and outer streaming vortices are visible. This result is also found by Hamilton et al. [3]. For wider channels, y = 2δ ν, the transverse streaming anti-node location ( = ) approaches the value found by ayleigh of y = y / 3 []. Moreover, the streaming patterns are all symmetric around the horizontal anti-node location x = λ/4. In all four cases the calculated L 2 -norm is less than 5% of the peak streaming velocity u. Tube Geometry In Figure 3 the simulation results for the various tubular simulation cases T. T.4 are shown. The streaming vortices are clearly visible from the streamlines plots and a transition from inner streaming to both inner and outer streaming is observed between = 5δ ν and = δ ν. The calculated streaming patterns are symmetric around the horizontal anti-node location x = λ/4 in all four different cases. Proceedings of Meetings on Acoustics, Vol. 9, 38 (23) Page 6
7 L/2 L L/2 L L/2 L L/2 L (A) P.: y = 3δ ν (B) P.2: y = 5δ ν (C) P.3: y = δ ν (D) P.4: y = 2δ ν (E) for P.: y = 3δ ν (F) for P.2: y = 5δ ν (G) for P.3: y = δ ν (H) for P.4: y = 2δ ν.5 2 λ/2.5.5 λ/2.5 λ/2.5 λ/ (I) for P.: y = 3δ ν (J) for P.2: y = 5δ ν (K) for P.3: y = δ ν (L) for P.4: y = 2δ ν FIGUE 2: Simulation results of parallel plate geometry compared to analytical results for various half plate spacings y. First row shows streamline plot of simulated streaming velocity. Second row shows transverse distribution of x-component of streaming velocity at streaming anti-node location x = L/4. Third row shows transverse distribution of y-component of streaming velocity at streaming node location x = L/2. Solid line indicates simulation result, dotted line represents analytical solution. CONCLUSIONS From the quantitative comparison of the parallel plate simulation cases with the analytical solution of Hamilton et al. [9], it can be concluded that it is possible to simulate ayleigh streaming with commercial CFD software within an accuracy of 5%. This accuracy can be obtained using a simulation mesh consisting of only 6 mesh elements. However, for the calculation of smooth streaming velocity fields, a mesh resolution of N x = 44, N y = 3 was used which yields a total of 3,2 mesh elements. This study shows that the presented CFD model is suitable for quick design optimization and parameter sensitivity studies to reduce ayleigh streaming velocities and corresponding energy losses in standing waves. Similar results are obtained by using a tube geometry which is more common in current thermoacoustic devices. These results pave the way for the simulation of ayleigh streaming in more complex geometries, as well as at higher pressure amplitudes such as present in real thermoacoustic devices. Hence, this research provides a base for studies to reduce the negative effects ayleigh streaming can have on standing wave thermoacoustic prime mover and refrigerator efficiency. In the near future this work will be continued with the simulation of ayleigh streaming in a thermal buffer tube of a thermoacoustic prime mover instead of a closed resonator as described in this current paper. The effect of geometric changes on the streaming losses will be investigated and the simulation conditions will be changed to match the realistic operation conditions with Proceedings of Meetings on Acoustics, Vol. 9, 38 (23) Page 7
8 L/2 L L/2 L L/2 L L/2 L (A) T.: = 3δ ν (B) T.2: = 5δ ν (C) T.3: = δ ν (D) T.4: = 2δ ν (E) for T.: = 3δ ν (F) for T.2: = 5δ ν (G) for T.3: = δ ν (H) for T.4: = 2δ ν.5 2 λ/ λ/2.5 λ/2.5 λ/ (I) for T.: = 3δ ν (J) for T.2: = 5δ ν (K) for T.3: = δ ν (L) for T.4: = 2δ ν FIGUE 3: Simulation results of tube geometry for various radii. First row shows streamline plot of simulated streaming velocity. Second row shows radial distribution of x-component of streaming velocity at streaming anti-node location x = L/4. Third row shows transverse distribution of r-component of streaming velocity at streaming node location x = L/2. corresponding temperature boundary conditions and high pressure amplitudes. Moreover, further steps will be performed to predict the streaming induced loss in the thermal-to-acoustic efficiency of the thermoacoustic system under investigation. By characterizing the convective heat transport using the simulated ayleigh streaming velocity field, the loss in efficiency will be estimated. The aim is to further reduce this efficiency loss with CFD based design optimization. ACKNOWLEDGMENTS The authors would like to thank Agentschap NL for the financial support as part of the EOS-KTO program, project number KTOT39. EFEENCES [] Lord ayleigh, On the circulation of air observed in Kundt s tubes, and on some allied acoustical problems, Philosophical Transactions of the oyal Society of London 75, 2 (884). [2] S. Boluriaan and P. J. Morris, Acoustic streaming: from ayleigh to today, International Journal of Aeroacoustics 2, (23). Proceedings of Meetings on Acoustics, Vol. 9, 38 (23) Page 8
9 [3] M. F. Hamilton, Y. A. Ilinskii, and E. A. Zabolotskaya, Acoustic streaming generated by standing waves in two-dimensional channels of arbitrary width, The Journal of the Acoustical Society of America 3, 53 (23). [4] G. W. Swift, Thermoacoustics: A Unifying Perspective for Some Engines and efrigerators (Acoustical Society of America) (23). [5] J. Olson and G. Swift, Acoustic streaming in pulse tube refrigerators: Tapered pulse tubes, Cryogenics 37, (997). [6] C. Eckart, Vortices and streams caused by sound waves, Physical eview 73 (948). [7] P. Westervelt, The theory of steady rotational flow generated by a sound field, The Journal Of The Acoustical Society of America 25, 6 67 (953). [8] W. Nyborg, Acoustic Streaming due to Attenuated Plane Waves, The journal of the acoustical society of America 25, (953). [9] M. F. Hamilton, Y. A. Ilinskii, and E. A. Zabolotskaya, Thermal effects on acoustic streaming in standing waves, The Journal of the Acoustical Society of America 4, 392 (23). [] H. Bailliet, V. Gusev,. aspet, and. a. Hiller, Acoustic streaming in closed thermoacoustic devices, The Journal of the Acoustical Society of America, 88 (2). []. Waxler, Stationary velocity and pressure gradients in a thermoacoustic stack, The Journal of the Acoustical Society of America 9, 2739 (2). [2] S. Bühler, Discretization of thermoacoustic CFD Simulations, Technical eport, University of Twente, Enschede (22). [3] COMSOL, COMSOL Multiphysics eference Guide (COMSOL) (22). [4] P. Morris, S. Boluriaan, and C. Shieh, Numerical simulation of minor losses due to a sudden contraction and expansion in high amplitude acoustic resonators, Acta Acustica united with Acustica 9, (24). [5] L. Menguy and J. Gilbert, Non-linear acoustic streaming accompanying a plane stationary wave in a guide, Acta Acustica united with Acustica 86, (2). Proceedings of Meetings on Acoustics, Vol. 9, 38 (23) Page 9
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