Acoustic characteristics of bi-directional turbines for thermoacoustic generators

Abstract

Bi-directional turbines combined with rotary motors may be a feasible option for developing high power thermoacoustic generators with low cost. A general expression for the acoustic characteristics of the bi-directional turbine was proposed based on theoretical derivation, which was validated by computational fluid dynamics modeling of an impulse turbine with fixed guide vanes. The structure of the turbine was optimized primarily using steady flow with an efficiency of near 70% (the shaft power divided by the total energy consumed by the turbine). The turbine in the oscillating flow was treated in a lumped-parameter model to extract the acoustic impedance characteristics from the simulation results. The key acoustic impedance characteristic of the turbine was the resistance and inertance due to complex flow condition in the turbine, whereas the capacitance was treated as an adiabatic case because of the large-scale flow channel relative to the heat penetration depth. Correlations for the impedance were obtained from both theoretical predictions and numerical fittings. The good fit of the correlations shows that these characteristics are valid for describing the bi-directional turbine, providing the basis for optimization of the coupling between the thermoacoustic engine and the turbine using quasi-one-dimensional theory in the frequency domain.

Graphical abstract

Keywords

thermoacoustic power generator / acoustic characteristics / bi-directional impulse turbine / energy conversion

1 Introduction

Nowadays, due to environment and energy crisis, it is an urgent task to explore a green and economic energy technology. The thermoacoustic power generation technology provides a good solution for this problem because it has the advantage of providing an environmentally friendly working medium with high reliability, and can recover low-grade heat. During the last three decades, numerous research studies have shown great potential for thermoacoustic heat engine (TAHE) in refrigeration and electricity generation [1–5]. The thermoacoustic electrical generator (TAEG) is composed of a TAHE and an acoustic-to-electric converter, which are acoustically coupled. During the operation of the TAEG, the heat energy is converted, through the TAEG, into acoustic power which is then used by the acoustic-to-electric conversion device for power generation [6–8].

To date, the acoustic-to-electric converters that have been reported for the TAEG mainly include linear generators, magnetohydrodynamic generators (MHD), piezoelectric transducers, and crank-rod transducer (CRT). Backhaus et al. [3] first reported a traveling wave TAEG which used a linear generator as the acoustic-to-electric conversion device in 2004. A maximum electrical power of 58 W was finally obtained [3]. Subsequently, the efficiency and output power of the TAEG were improved by Luo’s research group [4,5]. However, in practice, the linear generators encountered many problems, including high cost, low specific power, and some serious technical difficulties, e.g., the clearance seal, machining accuracy besides the fact that the supporting technology is difficult to realize for high power [9]. In 2007, Castrejón-Pita and Huelsz proposed a TAEG based on MHDs [10]. However, the result was quite preliminary, and no more details were reported. Efforts regarding the use of a piezoelectric transducer for TAEG were reported by a number of research groups. Nevertheless, the piezoelectric transducer is only suitable for miniature applications due to its high resonance frequencies and low conversion efficiency [11,12]. In traditional Stirling engines, the crank connecting rod structure is used to convert acoustic power to rotary shaft power. In 2017, Luo et al. investigated the operating and impedance characteristics for TAEG with CRTs [13,14]. But due to the mechanical defects of crank-rod, e.g., lubricating oil leads to the pollution of working medium and dynamic seal leads to the leakage of working medium, it is also difficult to meet the conventional needs for the short maintenance cycle. So, exploring a high specific power, low cost, and high-efficiency device used for acoustic power conversion has become an important contribution to the further development of TAEG.

Bi-directional turbines combined with rotary motors provide a feasible alternative in circumstances requiring high power and low cost [9]. They are commonly used in oscillating water column (OWC) plants as wave energy converters [15–17]. The axial-flow impulse bi-directional turbine is widely used in OWC plants due to its large operational flow range. The blade is symmetrical as is the arrangement of guide vane rows. Flow from either side drives the blade to rotate in the same direction.

While the turbine design has provided a relatively mature understanding in OWC, their application for acoustic power conversion in thermoacoustic engines has just begun due to their different work environment. There are much higher frequencies (20–300 Hz) and higher pressures (1–15 MPa) in thermoacoustic engines than for OWC, for which the air oscillating frequency is below 1 Hz and the pressure is atmospheric. In OWC, air flows are mainly incompressible, whereas in thermoacoustic engines, a compressible flow of low Mach number dominates. Acoustic power is the main form of energy propagating among all of the components, and thereby strongly couples them together. This implies that a holistic modeling of the thermoacoustic system is required for optimization.

All components of the thermoacoustic engine basically follow the linear thermoacoustic theory in which the time-averaged energy conversion is mainly a product of first-order wave components. Nonlinear effects from flow friction and heat transfer are effectively combined in the first-order frequency-domain wave equations. A quasi-one-dimensional frequency-domain optimization is traditionally implemented when designing a thermoacoustic system, as, for example, in the general thermoacoustic design software DeltaEC [18] from Los Alamos National Laboratory and SAGE developed by David Gedeon. Due to the domination of the strongly nonlinear, multi-dimensional flow in bi-directional turbines, the time-domain design method must also be used. However, a complete computational fluid dynamics (CFD) model simulating the thermoacoustic engine driving the bi-directional turbine would be too time-consuming to be practical. A coupling gap between the thermoacoustic engine and the turbine must be bridged. One solution is to lump the acoustic characteristics of the turbine together into a thermoacoustic model. The question then rises whether there is a universal representation of a bi-directional turbine via its lumped acoustic characteristics. To date, there are only a few qualitative analysis and simple experiments in the relevant researches. Kees et al. tested several bi-directional turbines in TAHE since 2012 [19]. A 300 mm diameter bi-directional turbine was reported in a multi-stage 100 KWT TAEG with air at 1 MPa mean pressure. The measured efficiency defined therein as the mechanical shaft power over the acoustic power was 76%. But the details of the measurement were not available. In 2015, Kaneuchi and Nishimura used two loudspeakers to provide the acoustic field for the bi-directional turbine [20]. But only simple tests were presented and the output power was less than 1 W with an efficiency less than 12%. It can be seen from the above research that the optimization between the coupled turbine and thermoacoustic engine has not yet been accomplished, nor has a systematic study of the characteristics of the turbine under oscillating flows been conducted. In this paper, a general expression for the impedance characteristic of an arbitrary flow duct suitable for bi-directional turbines is derived. Then, with the help of CFD simulations, the basic flow features of the impulse bi-directional turbine with fixed guide vanes in oscillating flows is investigated. The lumped impedance of the turbine is extracted from the results of simulations given various conditions. The impedance expression of the turbine is verified, which helps in understanding the working mechanism, and providing preconditions when optimizing the coupled thermoacoustic engine and turbine using the thermoacoustic theory.

2 Theoretical basis

For characterizing a flow duct in acoustics, there are several equivalent ways [21,22]. In this paper, the impedance is defined as the complex amplitude of the pressure gradient divided by the complex volume flow rate amplitude. The key acoustic impedance characteristic of the turbine is the resistance and inertance due to the complex flow condition in the rotor blade and its guide vanes, whereas the capacitance can be normally treated as adiabatic case because of the large-scale flow channel relative to heat penetration depth. Therefore, the acoustic characteristic analysis of this paper focuses on the acoustic impedance of resistance and inertance.

2.1 Linear impedence

From the linear thermoacoustic theory, the governing relation for flows through the ducts of arbitrary cross section without a time-averaged temperature gradient is written as Eq. (1) [23].

(1){\displaystyle \frac{\mathrm{d}{P}_{\mathrm{1}}}{\mathrm{d}x}}=-\mathrm{i}\omega {\rho }_{\mathrm{0}}{F}_{\mu }^{-\mathrm{1}}\cdot \overline{u_{\mathrm{1}}},

where P₁ and u₁ are the first-order components of the pressure wave and velocity wave, respectively; ω denotes the angular frequency; r₀ denotes the time average density; and F_μ denotes the viscous function. This equation is deduced from the momentum equation in the frequency domain assuming small perturbations. The effect of viscosity of the tube wall is contained in the viscous function F_μ via the cross-sectional average and is a nondimensional complex function depending only on the duct structure and flow conditions. For a duct where the hydraulic radius is far larger than the viscous penetration depth, specifically, the dynamic Reynolds number Re_ω→∞, the acoustic waves undergo inviscid propagation. For Re_ω→0, viscous flow dominates and tends towards a steady state. When the dynamic Reynolds number approaches infinity or zero, the limit expressions for the viscous function with a higher precision are expressed in Eqs. (2) and (3) [24].

(2){{\displaystyle F}}_{\mu }^{-\mathrm{1}}\left({{\displaystyle \mathit{Re}}}_{\omega }\to \infty \right)=\mathrm{1}+{\displaystyle \frac{\mathrm{2}\sqrt{\mathrm{2}}}{\sqrt{{{\displaystyle \mathit{Re}}}_{\omega }}}}-\mathrm{i}{\displaystyle \frac{\mathrm{2}\sqrt{\mathrm{2}}}{\sqrt{{{\displaystyle \mathit{Re}}}_{\omega }}}},

(3){{\displaystyle F}}_{\mu }^{-\mathrm{1}}\left({{\displaystyle \mathit{Re}}}_{\omega }\to \mathrm{0}\right)=\mathrm{1}+{\displaystyle \frac{\mathrm{1}}{{{\displaystyle \alpha }}_{\mathrm{1}}}}-\mathrm{i}{\displaystyle \frac{\mathrm{4}{{\displaystyle \alpha }}_{\mathrm{2}}}{{{\displaystyle \mathit{Re}}}_{\omega }}},

where the Reynolds number is R{e}_{\omega }=\rho \omega {{\displaystyle d}}_{\mathrm{h}}^{\mathrm{2}}/\mu ,, d_h denotes the hydraulic diameter, m denotes the kinetic viscosity, and α₁ and α₂ are constants characterizing the cross section of the duct. For example, α₁ and α₂ are 3 and 8, respectively, for a circular tube. In the following derivation, it is shown that α₁ represents the influence of flow duct shape on inertia, and α₂ represents the influence of flow duct shape on resistance. Following Ma’s derivation [25] in obtaining the transitional expression for a circular tube over the full range of Re_ω, the general approximate expressions for the real and imaginary parts of the viscous function are obtained by using Eqs. (4) and (5).

(4)\mathrm{Real}\left[F_{\mu }^{-\mathrm{1}}\left(R{e}_{\omega }\right)\right]=\mathrm{1}+{\left[{\left(\sqrt{R{e}_{\omega }}/\mathrm{2}\sqrt{\mathrm{2}}\right)}^{\mathrm{2}}+{\alpha }_{\mathrm{1}}^{\mathrm{2}}\right]}^{-\mathrm{1}/\mathrm{2}},

(5)\mathrm{Im}\left[F_{\mu }^{-\mathrm{1}}\left(R{e}_{\omega }\right)\right]=-\sqrt{{\left(\mathrm{2}\sqrt{\mathrm{2}}/\sqrt{R{e}_{\omega }}\right)}^{\mathrm{2}}+{\left(\mathrm{4}{\alpha }_{\mathrm{2}}/R{e}_{\omega }\right)}^{\mathrm{2}}},

where Real represents the real part and Im represents the imaginary part. A final approximate expression is obtained for the viscous function by using

{{\displaystyle F}}_{\mu }^{-\mathrm{1}}\left({{\displaystyle \mathit{Re}}}_{\omega }\right)\approx \mathrm{1}+{{\displaystyle \left({{\displaystyle \alpha }}_{\mathrm{1}}^{\mathrm{2}}+{\displaystyle \frac{{{\displaystyle \mathit{Re}}}_{\omega }}{\mathrm{8}}}\right)}}^{-\mathrm{1}/\mathrm{2}}

(6)-\mathrm{i}{\displaystyle \frac{\mathrm{4}{{\displaystyle \alpha }}_{\mathrm{2}}}{{{\displaystyle \mathit{Re}}}_{\omega }}}{{\displaystyle \left(\mathrm{1}+{\displaystyle \frac{{{\displaystyle \mathit{Re}}}_{\omega }}{\mathrm{2}{{\displaystyle \alpha }}_{\mathrm{2}}^{\mathrm{2}}}}\right)}}^{-\mathrm{1}/\mathrm{2}}.

This approximation expression has an acceptable accuracy for circular cross-sectional tubes [24]. For other regular shapes, such as parallel plates, triangles, and rectangles, the results are similar. The importance of Eq. (6) is that it provides a general expression for complex duct structures where analytical expressions cannot be obtained, e.g., porous media regenerators and bi-directional turbines with rotating blades.

Considering the porosity ε and tortuosity q of the flow duct, the impedance Z based on the linear differential equation of momentum has a general form of

(7)Z\equiv -{\displaystyle \frac{\mathrm{d}{P}_{\mathrm{1}}}{\mathrm{d}x}}/{\overline{u}}_{\mathrm{1}}=\mathrm{i}\omega {\rho }_{\mathrm{0}}{\displaystyle \frac{q^{\mathrm{2}}}{\epsilon }}{F}_{\mu }^{-\mathrm{1}}=R_{\mathrm{f}}\mathrm{+i}\omega L,

where R_f denotes the fluid resistance, and L denotes the acoustic inertia. The porosity ε represents the cross-sectional area ratio of the fluid in the turbine to tube. The tortuosity q represents the ratio of the real fluid path in the turbine to the axial distance between the turbine inlet and outlet.

2.2 Nonlinear impedence

In the turbine, because of the variations in flow direction and path, the nonlinear effects of fluid inertia dominate, implying that the impedance of the turbine is nonlinear. For steady incoming flows, the Darcy friction factor f_D is defined by Eq. (8).

(8)f_{\mathrm{D}}\equiv \left(-\mathrm{d}P/\mathrm{d}x\right)d_{\mathrm{h}}/\left(\mathrm{0.5}{\rho }_{\mathrm{0}}{u}^{\mathrm{2}}\right)=\left(P_{\mathrm{in}}-P_{\mathrm{out}}\right)d_{\mathrm{h}}/\left(\mathrm{0.5}{\rho }_{\mathrm{0}}{u}^{\mathrm{2}}\mathrm{\Delta }x\right),

where d_h denotes the hydraulic radius; P denotes the pressure; u denotes the fluid velocity in steady flow; P_in and P_out denote the pressure at inlet and outlet, respectively; and Dx denotes the channel length. Similarly, the Darcy friction factor can be applied to oscillating flows,

(9)f_{\mathrm{D}}\equiv \left(-\mathrm{d}{P}_{\mathrm{1}}/\mathrm{d}x\right)d_{\mathrm{h}}/\left(\mathrm{0.5}{\rho }_{\mathrm{0}}\left|u_{\mathrm{1}}\right|u_{\mathrm{1}}\right),

substituting the pressure gradient of the oscillation by Eq. (7) yields

(10)f_{\mathrm{D}}R{e}_{\mathrm{1}}=\left(q^{\mathrm{2}}/\epsilon \right)\mathrm{2}\mathrm{i}R{e}_{\omega }{F}_{\mu }^{-\mathrm{1}},

where the Reynolds number for the oscillating flow is R{e}_{\mathrm{1}}=\rho |u_1|d_{\mathrm{h}}/\mu. Then, for steady flow, Re_ω→0, substituting F_μ by Eq. (6) yields

(11){{\displaystyle f}}_{\mathrm{D}}{{\displaystyle \mathit{Re}}}_{{{\displaystyle \mathrm{1}}}_{\left({Re}_{\omega }\to \mathrm{0}\right)}}\simeq \mathrm{8}{{\displaystyle \alpha }}_{\mathrm{2}}\cdot {\displaystyle \frac{q^{\mathrm{2}}}{\epsilon }}.

By simply replacing the steady-state limit with the nonlinear resistance, a simple nonlinear impedance expression, applicable to turbines, is obtained. Then, the nonlinear impedance of an arbitrary duct is expressed as

(12)Z\approx \omega {\rho }_{\mathrm{0}}{\displaystyle \frac{f_{\mathrm{D}}R{e}_{\mathrm{1}}}{\mathrm{2}R{e}_{\omega }}}{\left(\mathrm{1}+{\displaystyle \frac{R{e}_{\omega }}{\mathrm{2}{\alpha }_{\mathrm{2}}^{\mathrm{2}}}}\right)}^{\mathrm{1}/\mathrm{2}}+\mathrm{i}\omega {\rho }_{\mathrm{0}}{\displaystyle \frac{q^{\mathrm{2}}}{\epsilon }}\left[\mathrm{1}+{\left({\alpha }_{\mathrm{1}}^{\mathrm{2}}+{\displaystyle \frac{R{e}_{\omega }}{\mathrm{8}}}\right)}^{-\mathrm{1}/\mathrm{2}}\right].

By separating the real and imaginary parts of Z, the nondimensional resistance R_f/(ωρ₀) and nondimensional inertia L/ρ₀ can be obtained.

(13)R_{\mathrm{f}}/\left(\omega {\rho }_{\mathrm{0}}\right)={\displaystyle \frac{f_{\mathrm{D}}R{e}_{\mathrm{1}}}{\mathrm{2}R{e}_{\omega }}}{\left(\mathrm{1}+{\displaystyle \frac{R{e}_{\omega }}{\mathrm{2}{\alpha }_{\mathrm{2}}^{\mathrm{2}}}}\right)}^{\mathrm{1}/\mathrm{2}},

(14)L/{\rho }_{\mathrm{0}}={\displaystyle \frac{q^{\mathrm{2}}}{\epsilon }}\left[\mathrm{1}+{\left({\alpha }_{\mathrm{1}}^{\mathrm{2}}+{\displaystyle \frac{R{e}_{\omega }}{\mathrm{8}}}\right)}^{-\mathrm{1}/\mathrm{2}}\right],

from which inertia L\simeq {\rho }_{\mathrm{0}}\left(q^{\mathrm{2}}/\epsilon \right) can be derived for Re_ω→∞ and L\simeq {\rho }_{\mathrm{0}}\left(q^{\mathrm{2}}/\epsilon \right)\left(\mathrm{1}+{\alpha }_{\mathrm{1}}^{-\mathrm{1}}\right) for Re_ω→0. As α₁ is generally of order 10, the inductance is then nearly constant in general situations [24].

3 Bi-directional turbine CFD simulation

Simulations are proved to be an effective and efficient method to analyze a bi-directional turbine [26]. The two-dimensional (2D) simplification introduces deviations, mainly because the influence of the difference in radial velocity and tip clearance leakage are neglected. With the main objective being to find a way to characterize the acoustic impedance of the bi-directional turbine, optimizing the numerical calculation method and turbine structure are beyond the scope of this study. A one-stage impulse bi-directional turbine with fixed guide vanes (Fig. 1), previously optimized under steady-state inlet flow, was used as the model for CFD simulations.

3.1 Simulation model

For the 2D structure model, Fig. 2 shows the basic unit of the bi-directional turbine mid-plane with periodic boundaries and the details of the boundary mesh near the edge of the rotor blade. Although the rotor blade has an elliptical leading edge, the trailing edge prescribes a circle of fixed radius. The tip of the blade is rounded with a radius 0.5 mm. The guide vane is a circular arc without thickness. The outer diameter of the blade is 120 mm and the hub to tip ratio is 0.7. The number of rotor blades is 30, the same number as guide vanes. Hence, the width of the blade spacing is 10.68 mm. The turbine CFD model is extended upstream and downstream to 27.6 times the chord length. The structure was optimized previously using a steady inlet flow with the highest efficiency of 68.79% (flow coefficient= 3.5) at a Reynolds number of 283157 in helium at a 6 MPa pressure.

The model was analyzed using the commercial software Fluent^TM (19.0, ANSYS, Canonsburg, PA, USA). The solver used was a segregated transient solver, although a compressible model for oscillating pressurized helium was implemented. The solver was applicable as the Mach number was smaller than 0.1 in all scenarios. The pressure-velocity coupling was implemented using the PISO scheme, and for the spatial discretization of pressure, PRESTO was adopted. A second-order upwind method was applied for the momentum, density, and energy equations, while the second-order implicit method was applied for time discretization. For steady inlet flows, the time step size was set to have more than 50 points when the rotor blade moved through one blade spacing. For general oscillating inlet flows, the time step was set to ensure 1000 points in one period of flow oscillation and 50 points in one blade spacing for rotor blade. In viscous dominated instances, to calculate the inertia accurately, up to 5000 points in one oscillation period was used. Such fine time steps could also contribute to capture the transient time evolution for lift and drag forces.

The sliding mesh model was adopted to simulate the bi-directional turbine, both in steady flow and oscillating flow. The rotor and stator regions were separated by interfaces (Fig. 2). For steady inlet flows, the outlet and inlet boundaries were set as pressure outlet and mass flow inlet, respectively. For oscillating incompressible flows, the outlet and inlet boundaries were set as pressure and velocity, respectively, both oscillating sinusoidally with time. For oscillating compressible flows, the outlet and inlet boundaries were both set as active pistons with oscillating velocities. Dynamic mesh with a layering method was used to simulate the action of the pistons. Rotating periodic boundaries were used to allow a cascade effect on the CFD model. A constant mesh motion velocity was set for the rotor region to simulate the rotating speed of the blade.

An unstructured mesh was used in the domain around the turbine region, and a structured mesh was used near the inlet and outlet boundaries. The minimal number of mesh cells was 28268 (Fig. 2). Initial trials was conducted with mesh cells up to 51053 for both steady and 50 Hz oscillating inlet flows, whose results agree well with the previous describe meshes. Boundary layers with 5–10 layers were attached to all the turbine walls, and were refined when the oscillating frequency got higher as the viscous penetration depth got smaller. The mesh size on the interface was fixed at 0.485 mm, and the time step size was controlled to limit the displacement of the rotor region to within 0.485 mm at every time step. The mesh size in the flow direction of the inlet and outlet regions was fixed at 2 mm, and therefore the time step size was also limited by the piston displacement.

Water and helium were used as working fluids. The flow was set as fully turbulent, which means the flow experiences turbulence during the whole period when the velocity oscillates. Although there is no mature understanding of oscillating turbulent flows as yet, the Iguchi assumption (a turbulent flow at each instant in time is identical to that of fully developed steady flow) is normally used [27,28]. This means turbulence models can be used in steady flows to calculate the transient oscillating turbulent flow. Previous researchers were followed in choosing the standard k-ε model [26–28]. The energy model has to be turned on for compressible helium in oscillating flow scenarios as the ideal-gas model is needed. In this instance, a constant 300 K wall temperature was set for all walls of the turbine.

3.2 Data analysis

To obtain the performance, the model was extended to a three-dimensional (3D) annular channel containing 30 blades with an inner diameter of 84 mm and an outer diameter of 120 mm. Although only the forces at the mid-plane were calculated, they were treated the same for all blade sections and radii. For a turbine optimized using 2D simulations, the actual 3D efficiency is lower than the 2D result, as the angle between the fluid and the turbine at other sections deviates from the optimal setting at the mid-plane. In this paper, the 3D effect was ignored for simplicity. Hence, the moment of forces M, the output shaft power P_t, and turbine efficiency h for the whole turbine are

(15)M=N\cdot {\displaystyle {\int }_{{{\displaystyle R}}_{\mathrm{1}}}^{{{\displaystyle R}}_{\mathrm{2}}}{{\displaystyle F}}_{\mathrm{1}}}\cdot r\mathrm{d}r={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle NF}}_{\mathrm{1}}\left({{\displaystyle R}}_{\mathrm{2}}^{\mathrm{2}}-{{\displaystyle R}}_{\mathrm{1}}^{\mathrm{2}}\right),

(16){{\displaystyle P}}_{\mathrm{t}}=M{{\displaystyle \omega }}_{\mathrm{t}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle NF}}_{\mathrm{1}}\left({{\displaystyle R}}_{\mathrm{2}}^{\mathrm{2}}-{{\displaystyle R}}_{\mathrm{1}}^{\mathrm{2}}\right){{\displaystyle \omega }}_{\mathrm{t}},

(17)\eta ={\displaystyle \frac{P_{\mathrm{t}}}{H_{\mathrm{in}}-H_{\mathrm{out}}}},

where N denotes the blades number; F_l denotes the time-averaged lift force; R₁ and R₂ denote the hub radius and the tip radius, respectively; r denotes the microelement length; ω_t denotes the angular frequency of the turbine rotation; and H_in and H_out denote the inlet total energy and outlet total energy, respectively. For steady flows, the total energy consumed is characterized by the pressure drop across the turbine. For oscillating flows, the total energy consumed takes contributions from the acoustic power losses across the turbine.

For oscillating flows, an equivalent velocity u_e may be used to define the flow coefficient \mathrm{\Phi } and may be comparable with that for steady inlet flows. It is defined using the equivalent kinetic energy over a single cycle (Eq. (18)) [29].

(18)u_{\mathrm{e}}={\left\{{\displaystyle \frac{\mathrm{1}}{T}}{\displaystyle {\int }_{\mathrm{0}}^T{\left[\left|u_{\mathrm{1}}\right|\sin \hspace{0.17em}\left(\omega t\right)\right]}^{\mathrm{2}}\mathrm{d}t}\right\}}^{\mathrm{1}/\mathrm{2}}=\left|u_{\mathrm{1}}\right|/\sqrt{\mathrm{2}},

where T denotes the period of the oscillating flow. The analysis shows that u_e is effective in obtaining the force associated with lift for both compressible and incompressible flows. Then, the flow coefficient \mathrm{\Phi } could be defined as

(19)\mathrm{\Phi }={\displaystyle \frac{u_{\mathrm{e}}}{U_{\mathrm{r}}}},

where U_r is the circumferential velocity in the mean radius of the turbine.

For the turbine, the velocities in and around both the blade and guide vane regions are clearly nonuniform. Nonlinearity plays an important role in determining the acoustic characteristics. The harmonics may change the wave forms for pressure and velocity [27]. However, if the system is dominated by a single frequency, for example, in a resonant system, or the frequency is controlled by the inlet and outlet boundaries, then the energy is still mainly carried by the fundamental frequency. The concept of an impedance Z may still be useful in describing the acoustic field across the turbine. The purpose here is to obtain a lumped Z, representing the average amplitude for the pressure gradient given a velocity oscillation amplitude of 1 m/s. The influence of variables, including amplitude Reynolds number, dynamic Reynolds number, and flow coefficient, was investigated. The lumped Z is then calculated using Eq. (20).

(20)Z={\displaystyle \frac{P_{\mathrm{1},\hspace{0.17em}\mathrm{in}}-P_{\mathrm{1},\hspace{0.17em}\mathrm{out}}}{\mathrm{\Delta }x\cdot \left(u_{\mathrm{1},\hspace{0.17em}\mathrm{in}}+u_{\mathrm{1},\hspace{0.17em}\mathrm{out}}\right)/\mathrm{2}}},

where P_1,in and u_1,in denote the first-order components of the pressure wave and velocity wave at the inlet, respectively; and P_1,out and u_1,out denote the first-order components of the pressure wave and velocity wave at the outlet, respectively

4 Results and discussion

From the simulation results, the flow field features and the impulse bi-directional turbine performance in the oscillating flow were investigated first. Next, a general expression suitable for expressing the nonlinear resistance characteristics of the turbine was obtained by fitting the data obtained in various conditions. The acoustic impedance based on Eq. (20) was calculated for both incompressible and compressible oscillating flows. Data fittings for the nondimensional real and imaginary parts of the impedance were performed under conditions of different amplitudes of Reynolds number, dynamic Reynolds number, and flow coefficient. Equations (13) and (14) were then validated.

4.1 Flow field features and turbine performance

To show the flow field features of the bi-directional impulse turbine, the flows oscillating at 50 Hz in 6 MPa helium were studied in detail. The blade was given a rotational speed of 3000 r/min. The outlet and inlet boundaries were set as pistons, both oscillating at a velocity amplitude of 45 m/s but with a phase difference of 5 °. The output shaft power was 1740 W and the efficiency of the turbine was 58.9%. The phase difference between the inlet and outlet velocities is given by assuming a similar traveling-standing wave ratio (TSWR) as in the resonance tube of a looped thermoacoustic engine. Nevertheless, the TSWR may change if the turbine is coupled with the engine near the velocity peak; hence, the phase difference may not be correct as the turbine is coupled with a thermoacoustic engine. However, the acoustic characteristics based on the differential equation are independent of the specific acoustic field, although the performance does depend on the coupled acoustic field.

Figure 3 shows the velocity and pressure in one oscillation period. The velocity for a flow in and out of the turbine are almost the same, while the remarkable pressure difference between the inlet and outlet may be used to calculate the impedance of the turbine. The rough fluctuations in pressure oscillations are caused by the rotation of blades.

Figure 4(a) demonstrates the time evolution of the forces. It can be seen that the lift force changes periodically but the value is always greater than 0. The output power only takes contributions from the time-averaged lift force while the time-averaged drag force is near 0 because of the symmetry of the structure along the flow direction and the negligible difference in velocities coming into the turbine from both directions. Figure 4(b) illustrates the turbine efficiency and shaft power in both steady flow and oscillating flow. With the increases in the incoming flow oscillating frequency, the turbine efficiency decreases. The optimal flow coefficient also decreases with a higher incoming flow frequency. Note that the output shaft power only depends on the flow coefficient. The field analysis of the CFD results indicates that the incoming flow frequency has a great influence on the turbulence intensity of the flow field, but has little effect on the pressure distribution on the rotor blade which generates lift. Therefore, the higher frequency increases the loss of fluid energy and then reduces the turbine efficiency.

4.2 Resistance characteristics in steady flow

For a turbine running with fixed blade velocity, the steady flow friction factor has the form of f_D = A/Re + B, where A and B are constants. However, considering the change of blade velocity and flow coefficient, the more general form for the friction factor of a turbine in steady flow is

(21)f_{\mathrm{D}}={\displaystyle \frac{C_{\mathrm{1}}}{{Re}_{\mathrm{1}}}}+{\displaystyle \frac{C_{\mathrm{2}}}{\mathrm{\Phi }}}+C_{\mathrm{3}},

where C₁, C₂, and C₃ are constants depending only on the turbine structure. For the turbine structure (Fig. 2), C₁ = 1358.1, C₂ = 0.5372, and C₃ = 0.0331 (Fig. 5). Governed by linear viscosity, the accuracy of these constants decreases when Re₁ is small. Nevertheless, the purpose here is to study the main rules of the flow friction around the turbine flow where, normally, nonlinear resistance dominates.

4.3 Impedance characteristics in oscillating flow

A large range of parameter settings is covered by varying the working fluid, the rotating speed of the turbine, flow coefficient, and the frequency of the oscillating flow. The Reynolds number covers the range from 10⁴ to 10⁶, whereas the dynamic Reynolds number is in the range from 2 to 10⁶ and the flow coefficient ranges from 0.5 to 20. Water was chosen as the medium first when calculating, and a comparison was made between water and helium then. For various settings, the inertia and resistance parts of the lumped impedance defined by Eq. (20) were calculated.

With flows oscillating with time, inertia L exists. For inviscid flow, there is only inertia term, including the effects of acceleration and nonuniform velocity distributions, in balancing the pressure gradient. The inertia terms then have the form of L/ρ₀ = q²/ε for Re_ω→∞. When Re_ω→0, the inertia term has the form of L/{\rho }_{\mathrm{0}}\simeq \left(q^{\mathrm{2}}/\epsilon \right)\left(\mathrm{1}+{\alpha }_{\mathrm{1}}^{-\mathrm{1}}\right). In 2D cases, the porosity ε here is 0.891 and the tortuosity q is 1.297. So, L/{\rho }_{\mathrm{0}}\simeq q^{\mathrm{2}}/\epsilon \approx \mathrm{2}. Figure 6(a) shows L/ρ₀ for these two limits and conforms with the analysis. From Fig. 6(b), it is found that L/ρ₀ is not very sensitive to any variable, but has a relatively greater sensitivity when frequency decreases and exhibits its highest value for quasi-steady flows. However, for quasi-steady flows, inertia is then neglected. Inertia is more important at higher frequencies; at the same time, L/ρ₀ becomes less sensitive to both frequency and rotation speed. The flow coefficient is then the main factor to be considered. There is no strong fitting formula to describe the relationship between L/ρ₀ and Re₁, F, Re_ω, as the value changes little to separate all of these effects. For simplicity, a constant value of L/ρ₀ is proposed for a specific turbine. Figure 7 depicts that the nondimensional inertia of compressible flow also follows the rules of incompressible fluid, which means that the compressibility effect on L can be neglected.

From Fig. 6(a), it can be found that inertia calculation for extreme Re_ω suggests that q²/ε may have a value around 2. So, here, q²/ε = 2 was used to calculate Eq. (13). Then R_f/(ωρ₀) could be calculated for all cases by inputting the Re₁, F, Re_ω (Fig. 8), and comparing with the practical R_f/(ωρ₀).

The overall results validate the general expressions. Figure 9 presents the verification of the nondimensional resistance expression of water (incompressible) and helium (compressible), separately. At each point in Fig. 9, the horizontal coordinate represents the CFD value while the vertical coordinate represents the calculated value under the same working condition. The red line means they are equal. The difference between the data points and the red line represents the difference between the calculated value and the theoretical value. The overall results validate the general expressions. Figure 9 also shows that the fluid compressibility has no obvious influence on the fluid resistance. However, an obvious deviation is observed for small resistances region as linear resistance characteristics may play an important role, but a strong nonlinear friction factor was employed in Eq. (13).

5 Conclusions

The acoustic characteristics for bi-directional impulse turbines in oscillating flow were studied. A general correlation suitable for describing nonlinear impedance of the turbine was deduced from a theoretical analysis. The nonlinear resistance is mainly determined by the steady-state resistance characteristics, while the inertance characteristic is mainly determined by the linear acoustic characteristics. Lumped impedance Z was then obtained by analyzing the CFD simulation of a specific axial impulse bi-directional turbine. The validity of the correlation was checked. The effect of compressibility on the complex impedance Z may be omitted. Nonlinear resistance is the main characteristic of the turbine when working in the acoustic field. The nondimensional inertia has a value larger than 1, signifying that the turbine has an enhanced inertial effect compared with straight flow ducts and is nearly constant for a given turbine structure.

In summary, the method of obtaining impedance expression is of universal significance. For practical bi-directional turbines used in thermoacoustic engines, the lumped impedance is applicable for optimizing the coupling between the turbine and thermoacoustic engine.

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