Turbulence Reynolds Time Averaging Turbulence n Turbulence or

Turbulence Reynolds Time Averaging

Turbulence n Turbulence or turbulent flow is a flow regime characterized by chaotic property changes. n At very low speeds the flow is laminar, As the speed increases, at some point the transition is made to turbulent flow.

Characteristics of Turbulent Flow: 1. 2. 3. 4. 5. Fluctuations: (of P, V, T and heat transfer) Eddies: (eddies size varies continuously from shear layer thickness about 40 mm, to kolmogorov length scale about. 05 mm) Random Variations: in fluid properties. Self-sustaining motion: (produce new eddies to replace those lost by viscous dissipation) Mixing: (significantly higher than laminar flow)

Time Averaging n Eg: velocity component n At a particular point average of the time is defined as: - n Where T is chosen to be larger than any significant period of fluctuations in u.

Fluctuations n Fluctuation in u: can be defined as difference between u and its mean value: - n And thus by definition the mean of fluctuation must be zero. viz.

Root Mean Square Value n Thus to find the order of fluctuations we define its root mean square value. n The mean square value of fluctuation is: - n And thus the root mean square value is: -

Fluid properties in turbulent flow. n The fluid properties in turbulent flow can be written as: - n where x can be u, v, w, , p, T. n And as per our definition: - And n Assuming that mean motion is independent of time, the flow is termed ‘steady turbulent flow’

Rules for averaging n The following rules imply for the time averages: n For f and g as dependent variable and s being independent variable then: -

Reynolds Stress n Momentum transfer due to turbulent fluctuation of velocity. n Average momentum flux (for const density): - But: n Thus the flux in y direction of the x momentum is : - n Reynolds/ apparent Stress: - (additional shear stress in x dir on surface element perpendicular to y direction)

Basic Equations for Mean Motion of Turbulent flow (Incompressible flow) n Continuity Equation: - n For mean motion: - n and

Momentum Equation n In X- direction: - n For Mean motion: -

n Or in Stress-tensor form: - n In similar manner we can write eq’s in Y and Z direction and thus can define stress tensor due to turbulent velocity component as: -

n While the complete Stress are given by: - n and

Kinetic Energy n The kinetic energy equation: - n With n And for mean motion the equation is given by: - as given by equation 16. 16 and 16. 17 in handouts.