# 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.[^1] 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.*[^2]

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.[^3]
    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{width="380"}
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{width="480"}
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.[^4] A
    recent paper gives orders of magnitudes of velocities observed in
    this type of acoustic streaming [^5]

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

!Figure 3: Gedeon, Rayleigh and jet-driven
streaming{width="480"}

*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.[^6]
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~=P~0~/ρ~0~c~0~*. 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* [^7]

## References


[^1]: see video on <http://media.efluids.com/galleries/all?medium=749>

[^2]: Greg Swift,\"Thermoacoustics: A unifying perspective for some
    engines and refrigerators\", Condensed Matter and thermal Physics
    Group,Los Alamos National Laboratory, Forth edition, 1999.

[^3]: S. Boluriaan, P. J. Morris,\"Acoustic streaming: from Rayleigh to
    today\", International Journal of Aeroacoustics, 2 (3-4): 255-292,
    2003.

[^4]: O. V. Rudenko, S. I. Soluyan,\"Theoretical foundations of
    nonlinear acoustics\", Consultants Bureau, New York and London,
    1977.

[^5]:
        .

[^6]: S. Moreau, H. Bailliet, J. Valiere,\"Measurements of inner and
    outer streaming vortices in a standing waveguide using laser doppler
    velocimetry\", Journal of Acoustical Society of America, 123 (2):
    640-647, 2008.

[^7]: M. W. Thompson, A. A. Atchley,\"Simultaneous measurement of
    acoustic and streaming velocities in a standing wave using lase
    doppler anemometry\", Journal of Acoustical Society of America,
    117:1828-1838, 2005.
