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On the energy transfer to small disturbances in fluid flow (Part I)

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Summary

The “energy in a small disturbance” in a viscous compressible heat-conductive medium is defined as a positive definite quantity characterizing the mean level of fluctuation in the disturbance which, in the absence of heat transfer at the boundaries and of work done by boundary forces or body forces, and in the absence of heat and material sources, is a monotone non-increasing function of time. For small disturbances a quantity satisfying these requirements is found. When viscosity and heat conductivity are neglected, it reduces to the familiar acoustic energy in the theory of sound. Stability in the mean of such a fluid system can thus be discussed with reference to the growth and decay of the energy in the disturbance.

The effects of body forces, heat and material sources are discussed.Rayleigh's criterion for the stability of systems involving heat sources is derived and its limitations shown. Transfer of energy from a steady main stream to a disturbance is then examined, and the particular case of a parallel main stream is worked out in detail. The last analysis will be useful in the discussion of the mechanism of hydrodynamic instability for a viscous compressible heat-conductive flow. In addition to the work done by theReynold's stress, there is another major energy transfer term caused by the transport of entropy spots across layers of fluids of different mean temperature.

Zusammenfassung

Die “Energie in einer kleinen Störung” in einem zähen, kompressiblen und wärmeleitenden Medium wird als eine positiv definierte Größe eingeführt, welche die mittlere Schwankung in der Störung charakterisiert und bei Abwesenheit von Wärme- und Massequellen eine monoton nicht anwachsende Funktion der Zeit ist, falls kein Wärmeübergang an der Oberfläche stattfindet und keine Arbeit von den Oberflächen- und Volumskräften geleistet wird. Für kleine Störungen wird eine solche Funktion angegeben. Sie reduziert sich, wenn Zähigkeit und Wärmeleitung vernachlässigt werden, auf die bekannte akustische Energie in der Theorie des Schalles. Mit Hilfe des Anwachsens und Abnehmens der Störungsenergie kann dann die Stabilität im Mittel diskutiert werden.

Die Einflüsse der Volumskräfte und der Wärme- und Massequellen werden besprochen. DasRayleighsche Kriterium für die Stabilität eines Systems mit Wärmequellen wird hergeleitet und sein Geltungsbereich gezeigt. Der Energieübergang von der stationären Hauptströmung auf die Störungen wird untersucht und der Sonderfall der Parallelströmung im einzelnen ausgearbeitet. Letztere Untersuchung erscheint nützlich bei der Diskussion des Mechanismus der hydrodynamischen Instabilität in einer zähen, kompressiblen, wärmeleitenden Strömung. Zusätzlich zur Arbeit, die von denReynoldsschen Spannungen geleistet wird, gibt es noch ein weiteres wichtiges Energieübertragungsglied, das vom Transport von Entropienestern quer durch Flüssigkeitsschichten mit verschiedener mittlerer Temperatur herrührt.

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Abbreviations

a :

velocity of sound of the flow

\(\bar a\) :

velocity of sound in the main stream

a 0 :

sound speed in an undisturbed uniform medium

C p :

specific heat at constant pressure

C v :

specific heat at constant volume

d j :

defined by Eq. (33e)

E :

energy in a disturbance; see Eqs. (9) and (34)

e ij :

rate of strain tensor\( = \left( {\frac{{\partial u_i }}{{\partial x_j }} + \frac{{\partial u_j }}{{\partial x_i }}} \right)\)

F i :

component of body force per unit mass in thex i -direction

F x, F y :

component of body force per unit mass in thex-, y-direction, respectively

g :

gravitational acceleration

h :

a non-negative quantity; see Eq. (27)

h j :

defined by Eq. (33 g)

i, j, k :

indices: 1, 2, or 3; summation convention being used throughout

K :

coefficient of conductivity

\(\bar K\),K 0 :

K at\(T = \bar T\) andT=T 0, respectively

K′ :

\(K - \bar K\)

\(\bar K_1\) :

\(\left( {\frac{{dK}}{{dT}}} \right)_{T = \bar T}\)

M, m′ :

rate of mass production per unit volume

\(\vec n\) :

normal vector at the boundaries

p :

pressure of the flow

\(\bar p\) :

pressure of the main stream

p′ :

pressure in the disturbance

p 0 :

pressure in an undisturbed uniform medium

Q′ :

rate of heat release per unit volume

Q * :

defined by Eq. (43)

R :

gas constant

S :

entropy of flow

\(\bar S\) :

entropy of the main stream

S′ :

entropy in the disturbance

S 0 :

entropy in an undisturbed uniform medium

T :

temperature of flow

\(\bar T\) :

temperature of the main stream

T′ :

temperature in the disturbance

T 0 :

temperature in an undisturbed uniform medium

t :

time

u i :

component of velocity of flow in thex i -direction

\(\bar u_i\) :

component of velocity of the main stream in thex i -direction

u i ′:

component of velocity of the disturbance in thex i -direction

u :

component of velocity of flow in thex-direction

\(\bar u\) :

component of velocity of the main stream in thex-direction

u′ :

component of velocity of the disturbance in thex-direction

v′ :

component of velocity of the disturbance in they-direction

x i & (x, y):

Cartesian coordinates

γ:

C p /C v

δ ij :

Kronecker delta

Ψ:

viscous dissipation function\( = \left( {\frac{1}{2}\tau _{ij} e_{ij} } \right)\)

\(\bar \Phi\) :

\(\frac{1}{2}\bar \tau _{ij} \bar e_{ij} \geqslant 0\)

Ψ′:

\(\frac{1}{2}\tau _{ij} ^\prime e_{ij} ^\prime \geqslant 0\)

ϱ:

density of flow

\(\bar \varrho\) :

density of the main stream

ϱ′:

density of the disturbance

ϱ0 :

density in the undisturbed uniform medium

σ ij :

stress tensor in the flow field

\(\bar \sigma _{ij}\) :

stress tensor in the main stream

σ ij ′:

stress tensor in the disturbance, Eq. (4)

dσ:

an element of surface area

τ ij :

viscous stress tensor in the flow field

\(\bar \tau _{ij}\) :

viscous stress tensor in the main stream, Eq. (30 a)

τ ij ′:

viscous stress tensor in the disturbance, Eqs. (4) and (33 a)

τ ij *:

defined by Eq. (33 f)

dτ:

a volume element

μ:

coefficient of viscosity

\(\bar \mu\),μ 0 :

coefficient of viscosity at\(T = \bar T\) andT=T 0, respectively

\(\bar \mu _1\) :

\(\left( {\frac{{d\mu }}{{dT}}} \right)_{T = T}\)

λ:

defined by Eq. (24 a)

Bar “−”:

signifies that quantities to which it is attached are associated with the main stream, e. g.,\(\bar p\) denotes the main stream pressure

Prime “′”:

signifies that quantities to which it is attached are associated with the disturbance, e. g.,p′ denotes the pressure in the disturbance

Subscript “0”:

signifies that quantities to which it is attached are associated with the undisturbed uniform state, e. g.,p 0 denotes the undisturbed pressure

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Authors and Affiliations

  1. Department of Engineering and Applied Science, Yale University, 06520, New Haven, Conn., U. S. A.

    Boa-Teh Chu

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  1. Boa-Teh Chu
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Additional information

Part I of this paper originally appeared as a report prepared under U. S. Air Force Contract AF 18 (600)-1121 with the Johns Hopkins University.

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Chu, BT. On the energy transfer to small disturbances in fluid flow (Part I). Acta Mechanica 1, 215–234 (1965). https://doi.org/10.1007/BF01387235

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