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Discussing the concept of turbulence of fluid dynamics

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## Discussing the concept of turbulence of fluid dynamics

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### Tor Håkon SivertsenBioforsk Plant Health and Plant Protection, Hogskoleveien7, N‑1432 Aas (Norway); e-mail:thsivert@online.no

Discussing the concept of turbulence of fluid dynamics

The scientific principle used in meteorology ( an interpretation)

We start by classifying natural phenomena ( put them into classes and sub-classes) like

air, cloud, soil, atmosphere, vegetational cover, canopy, leaf etc.

Then we may attach measurable quantities to the phenomena, like mass, temperature, energy, leaf area index, momentum etc.

A graphical representation of aninterpretation of thescientific principle usedin meteorology

The scientific principle ( an interpretation)

Testing and operational use of a model is not merely considered testing the hypotheses, but it is considered looking at the scope of the way we are classifying nature, the definition and measurements of parameters and the content of the hypotheses most often containing certain physical ‘laws’ like the conservation of energy, conservation of mass, conservation of momentum.

I make an unusual statement:

Any physical and biological phenomenon contains a totality that includes time, space and consciousness

I am using the concept of parameterization in this way:

We are connecting measurable/ quantitative entities to the phenomena – and we call these entities parameters

This includes length of time and the spatial

coordinates as being parameters defined in

principle like other parameters.

Through the centuries the following parameters (and many many others) are developed:

Length of time t, the spatial coordinates(x,y,z) , temperature of the air, pressure of the air, relative humidity of the air, density of the air, wind velocity of the air and density of the air. We call such parameters macro properties of the gas-mixture called air, or the parcels of air.

A Documentation System for Parameters (a) Measured(b) In Models

- Name of the parameter
- Unit
- Definition
- Representative-ness of phenomena in model considered
- Representativeness for phenomena in other models

- Name of the parameter
- Unit
- Defintion
- Method(s) for making measurement
- Representativeness for certain phernomena

Macro physics

The macro model of the air we may conceptually describe as a parcel of air (the mass is not clearly defined, merely the relative mass, the density); and connected to this parcel we have the quantitative parameters.

We may then extend our model by connecting spatial and temporal coordinates to each parcel of air (by using f.ex. Cartesian coordinates x,y,z and the time coordinate t) These coordinates are in fact parameters connected to each parcel of air.

Two different mathematical systems have been developed for studying flow of parcels of fluid: The representation of Joseph Louise Lagrange, looking at tagged parcels of fluid, and the representation of Leonard Euler,looking at the parameter values of the fluid parcels as function of the spatial and temporal coordinates.

Micro physics

There also are developed models of the molecular physics of the air, looking at the movements of the molecules.

This may be considered a quite different world with quite different phenomena: Molecules, space, time

We may connect parameters, measurable quantities to the

phenomena of the microphysics. And through statistical physics macro properties of the air may be derived.

.

Micro physics

Examples of parameters connected to the microphysics of the air: Molecular mass, velocity of a molecule, momentum of a molecule, angular momentum of a molecule, spatial coordinates (x,y,z), temporal coordinate t.

An interesting feature in this is that the temporal and spatial coordinates of the macro-physics and the micro-physics should not be the same. We consider two quite different worlds.

Physical ‘laws’

The parameters of the macro state, we connect to certain ‘physical laws’ or preliminary hypotheses containing combination of the parameters:

Conservation of mass

Conservation of energy ( containing The first law of thermodynamics).

Conservation of momentum

The second law of thermodynamics giving us the direction of certain processes.

Physical ‘laws’

We are able to use the laws of classical thermodynamics (the concept of reversible processes) for the parcels of air.

We are able to use Newtons laws of motion for each parcel of air (and we call this convective flow).

We then have a fluid model system containing two different interrelated processes going on simultaneously on two different scales, the molecular movements of the air and the convective movements of the air

What should a classification (definition) of the phenomenon of turbulence contain ?

(a) What do we mean by a classification of the phenomenon of turbulence?

(b) We want to connect quantitative(measurable) parameters to the phenomenon in order to derive mathematical/ physical expressions giving us fluxes of momentum, fluxes of heat, fluxes of latent heat etc.

Classification/ definition of turbulence?

(a) Obukhov: Turbulence cannot be defined (as a phenomenon)!

(b)Tennekes & Lumley and Goodarz Ahmadi): Turbulence is characterized by:

It is chaotic and seemingly random. It is highly diffusive. It is rotational and 3-dimensional motion. It has high levels of vorticity fluctuation-and vortex streching. It is highly dissipative. It is a continuum phenomenon much greater than molecular scale.It is characterized by high Reynolds-numbers. It is a manifestation of the flow and not of the fluid.

The mean field fluid is non-Newtonian, viscoelastic, memory-dependent, multitemperature, nonlocal, and contains several internal variables.

(c ) Osborn Reynolds did define turbulence as a phenomenon as chaotic movements in fluids different from laminar flow in fluids.

Is there a definition of turbulence contained in the Reynolds equations?

We use the Reynolds system to define a mean flow –

or a smooth flow system. The flow on smaller scale than the

mean flow we may call turbulence.( We could find another name for this). Also small scale stable gravity (internal) waves are then considered as ‘turbulence’.

The mean flow is then characterized by parcels of fluid much greater ( in temporal and spatial sense) than the parcels of fluid describing the Navier-Stokes flow system. The technical arrangements proposed by Reynolds for making averages, and interchanging this with differentiation is not always convincing, but systems for making averages should be considered even if differentiation procedures have to be redefined.

The total flow system may be characterized by the Navier-Stokes equations. The temporal and spatial coordinates of the two flow systems are not the same; but you can use continuum mechanics on both levels.

The use of the averaging procedure

The use of the procedure for averaging( temoral and spatial) is crucial.

It is in a way arbitrary, connected to the actual choice of the procedure.

The mean field fluid is non-Newtonian, viscoelastic, memory-dependent, multitemperature, nonlocal, and contains several internal variables.

Most procedures for making measurements of wind velocity contains also implicitly averaging compared to the wind velocity in systems defined by the Navier-Stokes equations.

In agro meteorology we usually are working with parameters like hourly averages of air temperature, hourly wind velocity 2m above the ground, hourly averages of relaive humidiy of the air etc.

Just a comment on the Reynolds number and parameters connected to studies of scale in turbulence

Each parameter has a ‘name’, it has a ‘definition’ and it has a ‘unit’. It is my opinion that all parameters used in a study generally should be explicitly defined in each case, also scaling parameters of temporal and spatial scale. The Reynolds number is usually defined in connection to stationary flows.

Looking at a viscous incompressible fluid, the Reynolds number Re=Re(x,y,z,t) may be considered a continuous function, telling something about the relation of acceleration to viscous resistance.

A comment on stationary systems of fluid flow

A lot of studies on turbulence are connected to stationary situations of the flow. Examples are stationary flow systems of air above a wide flat surface of ground. In situations with neutral stability of the air we get a logarithmic profile above the ground of the mean wind velocity field.

Also the famous studies of Monin and Obukhov on stable and unstable stratifications of air above the ground I will mention.

What is interesting is that the turbulent fluxes in these studies merely are functions of parameters contained in the average flow system.

This will never be the case in situations with change of mean flow system (accelerations) in time.

A last comment on energy budgets

We ought to arrive at a formal theoretical situation where the production on turbulent energy from the average flow system equals the viscous dissipation of turbulent energy into the world of molecular movements.

Is it of any value to use this as postulate if trying to formally construct turbulent eddies in another temporal and spatial coordinate system than the coordinate system of the average flow.

Then we will have no viscous dissipation in the mean flow system?

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