Predictions on momentum,heat and mass transfer in turbulent channel flow with the aid of a boundary layer growth-breakdown model

This paper deals with theoretical aspects of momentum, heat and mass transfer in turbulent channel flow and in particular with phenomena occurring close to the wall. The analysis presented involves the use of a boundary-layer growth-breakdown model. Theoretical expressions have been derived predicting heat and mass transfer at smooth surfaces in the fully developed and entrance region and at surfaces provided with ideal two-dimensional roughness elements. The analysis is restricted to fluids having Prandtl and Schmidt numbers larger than one. Good agreement appears to exist between theoretical predictions and experimental observations.

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Zusammenfassung Diese Arbeit behandelt die Theorie der Übertragungsvorgänge von Impuls, Wärme und Stoff in turbulenter Kanalströmung unter besonderer Berücksichtigung der Vorgänge in Wandnähe. Das verwendete Modell beruht auf dem Zusammenbruch der anwachsenden Grenzschicht. Für die ausgebildete Strömung und für den Einlaufbereich bei glatter Wand und bei Oberflächen mit idealen zweidimensionalen Rauhigkeitselementen werden theoretische Ausdrücke abgeleitet bei Beschränkung auf Prandtl- und Schmidt-Zahlen über Eins. Zwischen den theoretischen Voraussagen und den Versuchsergebnissen scheint gute Übereinstimmung zu herrschen.

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Nomenclature a thermal diffusivity m²/s] - c concentration kg/m³] - c _p specific heat J/kg °C] - D molecular diffusivity m²/s] - G relative increase in friction factor due to surface roughening - d pipe diameter m] - e height (depth) of roughness element m] - e ^p+ dimensionless roughness height (depth) - F parameter denoting the ratio - f friction factor for smooth surface and isothermal conditions - f _h friction factor for heating conditions - f _r friction factor for artificially roughened surface - n _av average frequency of fluctuations at the wall s^–1] - q heat flux W/m₂] - q _w heat flux at the wall W/m²] - q _wr heat flux at roughened wall W/m²] - q _wx wall heat flux to growing laminar boundary layer at positionx W/m²] - R _ma longitudinal correlation coefficient for mass transfer - R _mo longitudinal correlation coefficient for momentum transfer - T temperature °C] - T _b bulk temperature of fluid °C] - T ₀ fluid temperature at edge of viscous boundary layer (edge of viscous region) °C] - T _w wall temperature °C] - T _wx wall temperature at positionx for growing laminar boundary layer °C] - t time s] - t ₀ characteristic time period associated with boundary layer growth s] - u local axial fluid velocity, at wall distancey, for turbulent flow also denoting the mean velocity at that distance m/s] - u _b bulk fluid velocity m/s] - u ₀ fluid velocity at edge of viscous boundary layer (edge of viscous region) m/s] - u _0r fluid velocity at edge of viscous region for the case of an artificially roughened wall m/s] - u axial fluid velocity fluctuation m/s] - u ⁺ dimengionless fluid velocity,u/(_w/)^1/2 - u _i ⁺ instantaneous value ofu ⁺ - u _min ⁺ minimum value ofu _i ⁺ - u _r ⁺ root mean square value of dimensionless axial velocity - u ₀ ⁺ value ofu ⁺ at edge of viscous region - v fluid velocity normal to flow direction and normal to wall m/s] - v fluctuation of the velocityv m/s] - x coordinate in flow direction m] - x axial distance interval m] - x ⁺ dimensionless distance interval - x ₀ viscous boundary layer growth length m] - x ₀ ⁺ dimensionless boundary growth length - x _r axial dixtance between roughness elements m] - x _r ⁺ dimensionless distance between roughness elements - x _h value of viscous boundary growth length for heating conditions m] - y distance from wall m] - y ⁺ dimensionless wall distance - y _v thickness of viscous region m] - y _v ⁺ dimensionless form ofy _v - z _u unheated (zero mass transfer) part of elementary viscous boundary layer in entrance region m] - z _h heated (mass transfer) part of elementary viscous boundary layer m] - z _v lateral extent of elementary viscous boundary layer m] Greek symbols heat transfer coefficient defined with respect to bulk fluid temperature W/m² °C] - ₀ viscous region heat transfer coefficient W/m² °C] - _0h viscous boundary layer heat transfer coefficient averaged over lengthx ₀ for conditions of heating W/m² °C] - _0hh viscous region heat transfer coefficient averaged over lengthx _h for conditions of heating W/m² °C] - entrance region heat transfer coefficient at position W/m² °C - _,t viscous boundary layer heat transfer coefficient at position and timet W/m² °C] - mass transfer coefficient m/s] - _av average value of mass transfer coefficient m/s] - _x mass transfer coefficient for viscous boundary layer at positionx m/s] - entrance region mass transfer coefficient at position m/s] - thickness of laminar (viscous) boundary layer evaluated atu=1/2u ₀ m] - _max maximum value of boundary layer thickness m] - _i turbulent diffusivity for momentum transfer m²/s] - _h turbulent diffusivity for heat transfer m²/s] - _m turbulent diffusivity for mass transfer m²/s] - turbulent intensity - thermal conductivity W/m °C] - kinematic viscosity m²/s] - ₀ value ofv at edge of viscous region m²/s] - _w value ofv at the wall m²/s] - density kg/m³] - shear stress N/m²] - _tx local value of wall shear stress associated with viscous boundary layer growth N/m²] - ₀ value of wall shear stress averaged over lengthx ₀ N/m²] - _0r value of ₀ for the case of an artificially roughened wall N/m²] - _0h value of ₀ for heating conditions N/m²] - _h value of wall shear stress for heating conditions, averaged over lengthx _h N/m²] - _w wall shear stress for conditions of turbulent flow N/m²] - _wh value of _w for heating conditions N/m²] - dimensionless axial distancex/x ₀ in extrance region Dimensionless numbers Nu Nusselt number (d/) - Nu _x Entrance region Nusselt number at axial positionx - Nu _h Nusselt number for heating conditions - Nu _r Nusselt number for the case of artificially roughened surface - Pr Prandtl number (v/a) - Re Reynolds number (d u _b/v) - Re _b Boundary layer Reynolds number (1/2 u ₀/v) - Re _ber Critical value ofRe _b - Sh Sherwood number (d/D) - Sh _x entrance region Sherwood number at axial positionx - Sc Schmidt number (v/D)