Lecture of : the Reynolds equations of turbulent motions

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JORDANIAN GERMAN WINTER ACCADMEY. Lecture of : the Reynolds equations of turbulent motions. Prepared by: Eng. Mohammad Hamasha Jordan University of Science &amp; Technology . Most of the research on turbulent –flow analysis is the past century has used the concept of time averaging.

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### Lecture of :the Reynolds equations of turbulent motions

Prepared by:

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
• 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.

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:

equ….. 1

• From the basic integral Equation

incompressible continuity equation:

equ……...2

• Substitute in u, v, w from equ 1 and take the time average of entire equation

equ ……….3

• This is Reynolds-averaged basic differential equation for turbulent mean

continuity.

• Subtract equ 3 from equ 2 but do not take time average, this gives

equ……4

( compressible fluid ).

• Now: use non linear Navier-Stokes equation:

equ…5

• Convective- acceleration term:

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.

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.

rearranged to display the turbulent inertia terms as if they

• were stresses, which of course they are not. Thus we write:

equ…..8

• This is Reynolds-averaged basic differential equation for turbulent mean

momentum.

equ…..9

Turbulent

Laminar

• 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.

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

• This is Reynolds-averaged basic differential equation for turbulent mean

thermal energy.

equ…..12

equ…..13

Turbulent

Laminar

have collected conduction and turbulent convection terms into a

sort of total-heat-flux vector qi 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

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

II

III

I

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.