Lecture of : the Reynolds equations of turbulent motions JORDANIAN GERMAN WINTER ACCADMEY Prepared by: Eng. Mohammad Hamasha Jordan University of Science & Technology

Most of the research on turbulent –flow analysis is the past century has used the concept of time averaging. Applying time averaging to the basic equations of motion yield Reynolds equations. Reynolds equations involve both mean and fluctuating quantities. Reynolds equations attempt to model fluctuating terms by relating them to the mean properties or gradients. Reynolds equations form the basis of the most engineering analyses of turbulent flow.

assume :  Fluid is in a randomly unsteady turbulent state.  Worked with the time-averaged or mean equations of motion. So any variable is resolve into mean value plus fluctuating value  Where T is large compared to relevant period of fluctuation.

 Itself may vary slowly with time as the following figure fig 1

Lets assume the turbulent flow is incompressible flow with constant transport properties with significant fluctuations in velocity, pressure and temperature.  The variables will be formed as following: From the basic integral Equation equ….. 1

incompressible continuity equation: Substitute in u, v, w from equ 1 and take the time average of entire equation equ ……….3 Subtract equ 3 from equ 2 but do not take time average, this gives equ……...2 equ……4 This is Reynolds-averaged basic differential equation for turbulent mean continuity.

 equations 3 and 4 does not valid for fluid with density fluctuating ( compressible fluid ).  Now: use non linear Navier-Stokes equation:  Convective- acceleration term: equ…5 equ….6

 substitute equ 2 in to equ 5 equ…..7  the momentum equation is complicated by new term involving the turbulent inertia tensor.  This new term is never negligible in any turbulent flow and is the source of our analytic difficulties.  time-averaging procedure has introduced nine new variables (the tensor components) which can be defined only through (unavailable) knowledge of the detailed turbulent structure.

 The components of are related not only to fluid physical properties but also to local flow conditions (velocity, geometry, surface roughness, and upstream history).  there is no further physical laws are available to resolve this dilemma.  Some empirical approaches have been quite successful, though rather thinly formulated from nonrigorous postulates.

 A slight amount of illumination is thrown upon Eq. 7 if it is rearranged to display the turbulent inertia terms as if they were stresses, which of course they are not. Thus we write: Laminar Turbulent  mathematically, then, the turbulent inertia terms behave as if the total stress on the system were composed of the Newtonian Viscous Stresses plus an additional or apparent turbulent- stress tensor.  is called turbulent shear. equ…..8 equ…..9 This is Reynolds-averaged basic differential equation for turbulent mean momentum.

 Now consider the energy equation (first law of thermodynamics) for incompressible flow with constant properties equ…..10  Taking the time average, we obtain the mean-energy equation equ…..11 equ…..12 equ…..13 Laminar Turbulent This is Reynolds-averaged basic differential equation for turbulent mean thermal energy.

 By analogy with our rearrangement of the momentum equation, we have collected conduction and turbulent convection terms into a sort of total-heat-flux vector q i which includes molecular flux plus the turbulent flux.  The total-dissipation term is obviously complex in the general case. In two-dimensional turbulent-boundary-layer flow (the most common situation), the dissipation reduces approximately to equ…..14  Reynolds equations can not be achieve without additional relation or empirical modeling ideas

The Turbulence Kinetic-Energy Equations

 Many attempts have been made to add "turbulence conservation“ relations to the time-averaged continuity, momentum, and energy equations.  the most obvious single addition would be a relation for the turbulence kinetic energy K of the fluctuations, defined by equ….15

 A conservation relation for K can be derived by forming the mechanical energy equation, i.e., the clot product of u; and the ith momentum equation. then, we subtract the instantaneous mechanical energy equation from its time-averaged value. The result is the turbulence kinetic-energy relation for an incompressible fluid: equ….16

We have labeled this relation with Roman numerals to state the relation in words: The rate of change (I) of turbulent energy is equal to (II) its convective diffusion, plus (III) its production, plus (IV) the work done by turbulent viscous stresses, plus (V) the turbulent viscous dissipation. The terms in this relation are so complex that they cannot be computed from first principles. Therefore modeling ideas are needed. 

The Reynolds stress equation

From equation (8), the turbulent or "Reynolds" stresses have the form S ij =( ) From this point of view, the turbulence kinetic energy is actually Proportional to the sum of the three turbulent normal stresses, K = - S ii /2ρ. Of more importance to the engineer are the turbulent shear stresses, where i ≠ j. It is possible to develop a conservation equation for a single Reynolds stress. the derivation involves subtracting the time averaged moment equation (7), this yields Reynolds stress equation I IIIII IV V

 Here the roman numerals denote (I) rate of change of Reynolds stress, (II) generation of stress, (III) dissipation, (IV) pressure strain effects, and (V) diffusion of Reynolds stress.

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