An approximate theoretical analysis and experimental verification of turbulent entrance region flow of drag reducing fluids

Summary Entry lengths for pipe flows of moderately drag reducing fluids are determined using momentum integral technique. It is shown theoretically that the entry lengths for drag reducing fluids could be significantly larger than the Newtonian fluids flowing turbulently under otherwise identical conditions. The experimental data from the literature bear out the theoretical calculations.

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Zusammenfassung Mit Hilfe der Impuls-Methode wird die Einlauflänge in einer Rohrströmung für Flüssigkeiten mit mäßig starker Widerstandsverminderung berechnet. Es wird vorausgesagt, daß die Einlauflänge für derartige Flüssigkeiten erheblich größer sein kann als für newtonsche Flüssigkeiten unter sonst identischen Bedingungen. Aus der Literatur entnommene experimentelle Daten bestätigen diese theoretischen Berechnungen.

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Nomenclature A ₁ Coefficient in eq. 7] - A Slope of logarithmic velocity profile - a Exponent in eq. 10] - B Intercept function for logarithmic velocity profile - De Deborah number, - f Friction factor - F Function, eq. 30] - G Function given in eq. 11] - Static pressure, dynes/cm² - q Index of power law velocity profile - R Pipe radius, cm - r Radial distance, cm - R Core radius, cm - Re Reynolds number - Axial velocity, cm/s - u _c Core velocity, cm/s - u ⁺ Dimensionless velocity, eq. 5] - u ^* Friction velocity, , cm/s - Radial velocity, cm/s - V Average velocity, cm/s - x Axial distance, cm - x _e Entry length, cm - y Distance from the wall, cm - y ⁺ Dimensionless distance, eq. 5] - y _I ⁺ Dimensionless viscous sublayer thickness - coefficient in eq. 17] - exponent of Reynolds number in eq. 17] - shear rate, s^–1 - boundary layer thickness, cm - _fl fluid relaxation time, s - µ fluid viscosity, gm/cm s - v kinematic viscosity, cm²/s - _l laminar sublayer thickness, dimensionless - fluid density, gm/cm³ - shear stress, dynes/cm² - _w shear stress at the wall, dynes/cm² - ₁, ₂, ₃, ₄ functions in eq. 27] - ~ time averaged quantities - — dimensionless quantity With 3 figures and 1 table