TURBULENCE. Turbulence is highly diffusive. Rapid mixing significantly increases momentum, heat, and mass transfer.

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Brittney Bailey
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1 TRBLENCE Featres o Trblence i) Trblence is irreglar an seemingly ranom (chaotic). Statistical methos shol be se or extracting sel engineering inormation. ii) iii) iv) Trblence is highly isive. Rapi mixing signiicantly increases momentm, heat, an mass transer. Trblence is rotational an three-imensional motion. Trblence is associate with high levels o vorticity lctation. Smaller scales are generate by the vortex stretching mechanism. v) Trblence is highly issipative. It nees a sorce o energy to be maintaine. vi) vii) viii) Trblence is a continm phenomenon. The smallest scale o trblence is mch larger than the moleclar scales in most engineering applications. Trblence is a maniestation o low an not o the li. The mean iel li is non-newtonian, viscoelastic, memory-epenent, mltitemperatre, nonlocal, an contains several internal variables. Origin o Trblence Trblence is associate with high ynols nmber. Its origin is roote in the instability o shear lows. Trblence is also generate in boyancy riven lows. Energy Spectrm o Trblence Trblence has a wie range o length (time) scales. A typical energy spectrm (Forier ecomposition o energy) is shown in the igre. Here E () is the energy spectrm an is wave nmber (inverse wavelength ( l )). Flctation energy is proce at the large eies (with low wave nmbers). Vortex stretching mechanism E() niversal Eqilibrim Schematics o trblence energy spectrm., Inertia Sbrange, ME67

2 then generates smaller an smaller eies an energy lows own the spectrm to high wave nmber region. The energy is mainly issipate into heat at the smallest eies (o the orer o the Kolmogorov scales). The issipation rate, ε, is roghly eqal to the lctation energy proction rate. Sppose the large-scale velocity lctation o trblence is an the corresponing length scale is. Then the rate o proction (or issipation) o lctation energy is given by ε. () Eqation () implies that large eies lose a signiicant raction o their energy in a time perio o. Note that the irect viscos issipation rate is y ~ () an the ratio y ε L arge Ey Direct Viscos Dissip. Trblence Dissipation Rate, () where () ν is a characteristic ynols nmber. Kolmogorov Scales Large-scale trblent motion is roghly inepenent o viscosity. The smallscale, however, is controlle by viscosity. The small-scale motions are also statistically inepenent o relative slow large-scale trblent lctations (an/or mean motions). Accoring to Kolmogorov (niversal Eqilibrim Theory), the small-scale trblence is in eqilibrim (inepenent o large-scale) an is controlle solely with ε an ν. sing imensional argments, Kolmogorov eine the length, time, an velocity scales o the smallest eies o trblence. These are ME67

3 ν η, ε ν τ, ( νε). () ε sing eqation (), rom () it ollows that η ~ τ, ~, ~. (7) For a issipation rate o W o water, η 0µ m. g Kolmogoroc Inertia Sbrange Spectrm For eies mch smaller than the energy containing eies an mch larger than issipative eies (o the orer o Kolmogorov scales), trblence is controlle solely by the issipation rate ε an the size o the ey ( ). In this sbrange, E() ~ ~ ε ( ) ~ ε () which is the amos -/ law o Kolmogorov. In the erivation o (), simple ε imensional argment is se an velocity scale o eies o size ( ), ~ is se. The process o energy cascae in trblence is smmarize by Richarson verse: "Big whirls have little whirls that ee on their velocity. Little whirls have lesser whirls, an so on to viscosity." Ey Freqency Range For a pipe cm in iameter with water lowing at abot 0. m/s ( ~ 0 ), the energy containing eies have a reqency o abot 0-0 Hz. The issipative eies are in the range o Hz. For ~ 0, the issipative eies get into range o Hz. ME67

4 The size o large eies in the core region is abot 0% o the pipe iameter (i.e. abot cm). Approximate Estimates or Scales o Trblence For a pipe o iameter, the energy containing eies are o the orer o e l 0.0 (9) with a reqency o e ~ 0 l e, (0) τ where 0 is the shear velocity an is the mean velocity in the pipe. The ρ Kolmogorov length scale is given by 0.7 η, () an / ν 0.06 () 0. η is the reqency o the Kolmogorov eies. The size o the most issipative eies is (abot η ) 0.7 l 0 () with a reqency o ( ) () / ν ME67

5 The largest eies in the pipe are o the orer o with a reqency o abot L 0.. Example For water lowing throgh a 0 mm iameter pipe at velocity o. m/s (ynols nmber 0 ), the corresponing scales o trblence are smmarize in the table. Table o ey size an reqencies. Eies Size Freqency Largest Eies mm. Hz Energy Containing Eies 0.6 mm 0 Hz Most Dissipative Eies 0. mm µ m 0 Hz Kolmogorov Eies 0.0 mm µ m 00 Hz ME67

Rainer Friedrich

Rainer Friedrich Rainer Frierich et al Rainer Frierich rfrierich@lrztme Holger Foysi Joern Sesterhenn FG Stroemngsmechanik Technical University Menchen Boltzmannstr 5 D-85748 Garching, Germany Trblent Momentm an Passive