Engineering Acoustics/Acoustic streaming

Definition

Streaming is a term to describe a steady time-averaged mass-flux density or velocity induced by oscillating acoustic waves in a fluid. Large amplitude sound propagation in a fluid may result in a steady motion of the fluid. This nonlinear phenomenon is known as acoustic streaming and may be theoretically described by quadratic convective terms in the governing equations of the fluid flow.

Applications

Acoustic streaming may be effective in enhancement of convective heat transfer, ultrasonic cleaning of contaminated surfaces, localized micro-mixing, development of micro-actuators such as micro-manipulators for small particles and micro-pumps for liquids.Engineering Acoustics/Acoustic Micro Pumps

Different types of acoustic streaming

There exist several ways to classify streaming.

A first classification of streaming is based on the mechanism by which streaming is generated.

1. Boundary-layer driven streaming: Flow driven by viscous stresses on boundaries and caused by boundary layer effects between a solid and a fluid. Boundary-layer driven streaming consists of two types of streaming which occur always together: outer and inner boundary-layer streaming.
   1. Outer boundary-layer streaming: Rayleigh analysed acoustic streaming when a standing wave is present between parallel plates and explained that the air motion is caused by a nonlinear second order effect. Rayleigh focussed his investigations on mean flows outside the boundary layer and his approach became since then the analytical tool for the study of acoustic streaming.
   2. Inner boundary-layer streaming: The study of inner boundary layer streaming was developed by Schlichting, who investigated an incompressible oscillating flow over a flat plate and calculated the two-dimensional streaming field inside the boundary layer. Figure 1 shows the inner and outer streaming in a channel. The length of such a cell is a quarter of the wave length.

      

      Figure 1: Schematic diagram of inner and outer streaming cells

2. Jet driven streaming: Periodic suction and ejection of a viscous fluid through an orifice or a change in cross section. The mechanism relies on the fact that a viscous fluid behaves quite differently during the suction and ejection periods. During the suction period the flow comes from all kind of directions, whilst during the ejection period a jet is produced. In Figure 2, the outflow and inflow patterns at the transition between a small tube and open space are shown. These two outflows can be regarded as the superposition of a broadly distributed oscillating flow and a time-averaged toroidal circulation.

   

   Figure 2: Outflow and inflow patterns

3. Gedeon streaming: Associated with travelling wave, as opposed to a standing wave as for the previous example. In boundary layer and jet driven streaming, there is no net mass transport. In travelling wave streaming, a non-zero net mass transport occurs due to the phase between the acoustic velocity and density. Travelling wave streaming in Stirling thermoacoustic engines and refrigerators is called Gedeon streaming or DC flow.
4. Eckart streaming: Eckart streaming or 'quartz wind' is generated by the dissipation of acoustic energy in the fluid. Although Eckart was not the first to observe "quartz wind", he was the first one who gave a mathematical analyses for it in 1948. A recent paper gives orders of magnitudes of velocities observed in this type of acoustic streaming

Figure 3 shows the schematic of Gedeon, Rayleigh and Jet-driven streaming.

Figure 3: Gedeon, Rayleigh and jet-driven streaming

The second classification is based on the relative magnitude of the acoustic wavelength to the characteristic length of induced vortical structures.

1. Fine scale: For the inner boundary-layer streaming (Schlichting streaming), the boundary-layer thickness is equal to the width of the vortices. So, it is part of the fine scale classification.
2. Comparable scale: For outer boundary-layer streaming (Rayleigh streaming) and jet driven streaming, the wavelength and the vortex size are comparable.
3. Large scale: Eckart streaming belongs to the large scale classification because the vortex length scale exceeds the acoustic wavelength.

The third classification is based on the magnitude of the streaming velocity.

1. Slow streaming: Slow streaming is when the streaming velocity is considerably smaller than the magnitude of the fluid velocity. In fact, streaming can be characterized by an appropriate Reynolds number, Re_NL, which compares inertia and viscosity and determines the degree to which the streaming velocity field is distorted. The case Re_NL<<1 corresponds to the slow streaming.
2. Fast streaming: Fast streaming is when the streaming velocity and fluid velocity are of the same magnitude. The case Re_NL>>1 is referred to fast streaming or nonlinear streaming. Most types of acoustic streaming are slow, only the jet driven streaming is considered as fast.

Axial and transverse components of the streaming velocity

Here, the outer boundary-layer streaming velocity characteristics obtained from the analytical solution of the linear wave equation are presented. The amplitude of the axial component of the acoustic velocity field in the linear case is given as,

$(1)\ u=u_{max}sin(2\pi x/\lambda )$

where u_max=P₀/ρ₀c₀. The axial component (u_st) and the transverse component (v_st) of the streaming velocity field are,

$(2)\ u_{st}={\frac {3}{8}}{\frac {u_{max}^{2}}{c}}(1-{\frac {2y^{2}}{(H/2)^{2}}})sin(\pi x/l)$

$(3)\ v_{st}=-{\frac {3}{8}}{\frac {u_{max}^{2}}{c}}{\frac {2\pi y}{\lambda }}(1-{\frac {2y^{2}}{(H/2)^{2}}})cos(\pi x/l)$

where (-H/2<y<H/2), H is the height of the tube and l=λ/4