Summary
We initiate a study of acoustic streaming in a stratified fluid. A set of equations describing nondimensional field quantities is first derived. By introducing successive approximations, we next derive equations for the primary and the secondary motion. These equations are briefly discussed.
An adequat set of equations for the first order motion is then solved. The motion is modified by the stratification, but the feature of the first order motion is not essentially different from that in the uniform-fluid case.
The situation for the secondary motion is somewhat different. For sufficiently strong stratifaction the secondary motion is zero when calculated inLagrangian coordinates. For weak stratification, the flow pattern is similar to that of the uniform case—but the solutions are apriori valid for a limited time only. To estimate this timeinterval, the initial—value problem is studied briefly. Some numerical examples are also presented.
Zusammenfassung
Wir studieren eine akustische Strömung in einer geschichteten Flüssigkeit. Zunächst wird ein Gleichungssystem für die dimensionslosen Feldgrößen hergeleitet. Durch sukzessive Approximation erhält man weiters Gleichungen für die Primär- und Sekundärbewegung. Diese Gleichungen werden kurz diskutiert.
Ein adäquates System von Gleichungen für die Bewegung erster Ordnung wird dann gelöst. Die Bewegung wird zwar durch die Schichtung modifiziert, ihre Art ist aber nicht wesentlich veschieden gegenüber dem Fall der ungeschichteten Flüssigkeit.
Die Situation für die Sekundärbewegung ist einigermaßen verschieden. Für hinreichend starke Schichtung ist die Sekundärbewegung inLagrangescher Darstellung null. Für schwache Schichtung ist das Strömungsbild ähnlich dem im ungeschichteten Fall — jedoch sind die Lösungen hier von vornherein nur für eine beschränkte Zeit gültig. Um dieses Zeitintervall abzuschätzen, wird das Anfangswert-problem kurz studiert. Einige numerische Beispiele werden angegeben.
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Abbreviations
- g :
-
Acceleration of gravity
- k :
-
Wave number
- J :
-
Nondim. parameter\(g\beta /\omega ^2\)
- 0:
-
Order of magnitude symbol
- p :
-
Pressure
- p 0 :
-
Pressure in equilibrium
- p′ :
-
Perturbation ofp
- P :
-
Nondim. pressure =\(\varepsilon p'\sqrt {\rho _0 } \upsilon _0 ^2\)
- t :
-
Nondim. time=τ ω
- u :
-
Nondim. velocity=u 1+εu 2+...
- U :
-
Nondim. velocity outside the boundary layer
- v :
-
Physical velocity=v 1+εv 2+...
- V :
-
Physical velocity=V 1+V 2+...
- v 0 :
-
velocity amplitude of the oscillating obstacle
- X, Y, Z :
-
Physical cartesian coordinates
- x, y :
-
Nondim. coordinates =\(X/\lambda _x ,Y/\lambda _y\)
- α:
-
Integration parameter=λ+νk 2
- \(\beta ,\bar \beta\) :
-
Defined by eq. (15)
- γ:
-
Nondim. parameter = β λ y
- δ:
-
Thickness of the viscous A. C. boundary layer (B. L.)
- ε:
-
Nondim. parameter =\(\upsilon _0 /\omega \lambda _x\)
- ν:
-
Nondim. parameter =\(k\sqrt {v/\omega }\)
- λ:
-
Wavelength of the plates vibrations=2 π/k
- λ x , λ y :
-
Typical lengths in theX andY directions
- ν:
-
Kinematic viscosity
- ν′:
-
Nondim. kinematic viscosity =\(v/\omega \lambda _y ^2\)
- ϱ:
-
Mass density = ϱ0+γ(εϱ1+ε2ϱ2+...)
- ϱ0 :
-
Density in equilibruim =\(\bar \rho _0 \exp ( - \beta Y)\)
- ϱ:
-
Perturbation of ϱ0
- \(\hat \rho\) :
-
Nondim. density =\(\rho \sqrt {\rho _0 }\)
- \(\bar \rho _0\) :
-
Equilibrium density atY=0
- σ:
-
Mass density = σ0+γ(σ1+σ2+...)
- ψ1 :
-
Stream function, eq. (35)
- ω:
-
Frequency of oscillating obstacle
- τ:
-
Time
- <>:
-
Time average
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Kildal, A. On the motion generated by a plate vibrating in a stratified fluid. Acta Mechanica 9, 78–104 (1970). https://doi.org/10.1007/BF01176611
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DOI: https://doi.org/10.1007/BF01176611